Free Math Worksheets

- Sine is positive above the x-axis; cosine is positive to the right of the y-axis. - Sine and cosine values always lie between \(-1\) and \(1\). - Use nearby axis angles such as \(0^\circ\), \(90^\circ\), \(180^\circ\), and \(270^\circ\) to estimate size.

1. The angle \(10^\circ\) is in Quadrant I and near the positive x-axis, so its positive y-coordinate is small. Thus, \(\sin(10^\circ)\approx0.17\). 2. The angle \(100^\circ\) is in Quadrant II and near the positive y-axis, so its x-coordinate is small and negative. Thus, \(\cos(100^\circ)\approx-0.17\). 3. The angle \(260^\circ\) is in Quadrant III and near the negative y-axis, so its y-coordinate is close to \(-1\). Thus, \(\sin(260^\circ)\approx-0.98\). 4. The angle \(350^\circ\) is in Quadrant IV and near the positive x-axis, so its cosine is close to \(1\). A cosine value cannot exceed \(1\), so the correct choice is \(0.98\).

a) \(0.17\) b) \(-0.17\) c) \(-0.98\) d) \(0.98\)

- Add or subtract multiples of \(2\pi\). - Reduce each angle to a familiar unit-circle angle. - Recall exact sine and cosine values at multiples of \(\frac{\pi}{2}\).

1. Add \(2\pi\) to \(-\pi\): \(\sin(-\pi)=\sin(\pi)=0\). 2. Subtract \(2\pi\) from \(\frac{7\pi}{2}\): \(\cos\left(\frac{7\pi}{2}\right)=\cos\left(\frac{3\pi}{2}\right)=0\). 3. Subtract \(4\pi\) from \(\frac{11\pi}{2}\): \(\sin\left(\frac{11\pi}{2}\right)=\sin\left(\frac{3\pi}{2}\right)=-1\). 4. The angle \(-4\pi\) is coterminal with \(0\), so \(\cos(-4\pi)=\cos(0)=1\).

a) \(0\) b) \(0\) c) \(-1\) d) \(1\)

- Use the period \(2\pi\). - Distinguish even and odd multiples of \(\pi\). - Recall cosine at integer multiples of \(\frac{\pi}{2}\).

1. Cosine has period \(2\pi\). 2. The angles \(7\pi\), \(-3\pi\), and \(15\pi\) are odd multiples of \(\pi\), so each has cosine \(-1\). 3. The angles \(8\pi\), \(-2\pi\), and \(20\pi\) are even multiples of \(\pi\), so each has cosine \(1\). 4. The angles \(4.5\pi\), \(-1.5\pi\), and \(12.5\pi\) are odd multiples of \(\frac{\pi}{2}\) that are not integer multiples of \(\pi\), so each has cosine \(0\).

Cosine \(-1\): \(7\pi, -3\pi, 15\pi\) Cosine \(1\): \(8\pi, -2\pi, 20\pi\) Cosine \(0\): \(4.5\pi, -1.5\pi, 12.5\pi\)

- An angle with a degree symbol must be entered in degree mode; a unitless angle is interpreted in radians. - Use degree mode for parts a) and d). - Use radian mode for parts b) and c). - Check the fourth decimal place to round to the nearest thousandth.

1. For a), use degree mode: \(\sin(115^\circ) \approx 0.906307\), so the result is \(0.906\). 2. For b), use radian mode: \(\sin(2.8) \approx 0.334988\), so the result is \(0.335\). 3. For c), use radian mode: \(\sin\left(\frac{5\pi}{3}\right) = -\frac{\sqrt{3}}{2} \approx -0.866025\), so the result is \(-0.866\). 4. For d), use degree mode: \(\sin(-45^\circ) = -\frac{\sqrt{2}}{2} \approx -0.707107\), so the result is \(-0.707\).

a) \(0.906\) b) \(0.335\) c) \(-0.866\) d) \(-0.707\)

- Make sure your calculator is in degree mode. - Recall the angle intervals for the four quadrants. - Use the signs of the x- and y-coordinates on the unit circle to check the signs of cosine and sine.

1. For a), \(90^\circ < 142.5^\circ < 180^\circ\), so the angle is in Quadrant II. A calculator gives \(\sin(142.5^\circ)\approx0.6088\). 2. For b), \(180^\circ < 195^\circ < 270^\circ\), so the angle is in Quadrant III. A calculator gives \(\cos(195^\circ)\approx-0.9659\). 3. For c), \(270^\circ < 280.4^\circ < 360^\circ\), so the angle is in Quadrant IV. A calculator gives \(\sin(280.4^\circ)\approx-0.9836\). 4. For d), \(270^\circ < 340^\circ < 360^\circ\), so the angle is in Quadrant IV. A calculator gives \(\cos(340^\circ)\approx0.9397\).

a) \(\sin(142.5^\circ)\approx0.6088\); Quadrant II b) \(\cos(195^\circ)\approx-0.9659\); Quadrant III c) \(\sin(280.4^\circ)\approx-0.9836\); Quadrant IV d) \(\cos(340^\circ)\approx0.9397\); Quadrant IV

- A full rotation is \(2\pi\) radians. - Add or subtract multiples of \(2\pi\). - Check that one result lies in the required negative interval.

1. Coterminal angles differ by integer multiples of \(2\pi\). 2. Subtract \(2\pi\): \(\frac{3\pi}{4}-2\pi=-\frac{5\pi}{4}\), which lies in \([-2\pi, 0)\). 3. Add \(2\pi\): \(\frac{3\pi}{4}+2\pi=\frac{11\pi}{4}\). 4. Add \(4\pi\): \(\frac{3\pi}{4}+4\pi=\frac{19\pi}{4}\).

For example, \(-\frac{5\pi}{4}\), \(\frac{11\pi}{4}\), and \(\frac{19\pi}{4}\)

- Subtract full rotations of \(360^\circ\). - Identify the quadrant of the reduced angle. - Use exact special-angle values.

1. Reduce \(480^\circ\) to \(120^\circ\). Therefore, \(\sin(480^\circ)=\sin(120^\circ)=\frac{\sqrt{3}}{2}\) and \(\cos(480^\circ)=\cos(120^\circ)=-\frac{1}{2}\). 2. Reduce \(1125^\circ\) by three full rotations: \(1125^\circ-3\cdot 360^\circ=45^\circ\). Therefore, \(\sin(1125^\circ)=\frac{\sqrt{2}}{2}\) and \(\cos(1125^\circ)=\frac{\sqrt{2}}{2}\).

a) \(\sin(480^\circ)=\frac{\sqrt{3}}{2}\); \(\cos(480^\circ)=-\frac{1}{2}\) b) \(\sin(1125^\circ)=\frac{\sqrt{2}}{2}\); \(\cos(1125^\circ)=\frac{\sqrt{2}}{2}\)

- Coterminal angles differ by an integer multiple of \(360^\circ\). - Reduce each angle to the interval \([0^\circ, 360^\circ)\). - Compare each reduced angle with \(210^\circ\). - Be careful when adding full rotations to negative angles.

1. Reduce each angle to \([0^\circ, 360^\circ)\). 2. \(570^\circ-360^\circ=210^\circ\), so it has the same terminal side. 3. \(-150^\circ+360^\circ=210^\circ\), so it has the same terminal side. 4. \(910^\circ-2\cdot 360^\circ=190^\circ\), so it has a different terminal side. 5. \(-510^\circ+2\cdot 360^\circ=210^\circ\), so it has the same terminal side.

a) Same terminal side b) Same terminal side c) Different terminal side; the coterminal angle is \(190^\circ\). d) Same terminal side

- Add or subtract multiples of \(360^\circ\) to place each angle in the standard interval. - Evaluate the trigonometric function at the reduced angle and compare it with the original expression. - A negative angle represents clockwise rotation, but coterminal angles have the same terminal side.

1. \(980^\circ-2\cdot 360^\circ=260^\circ\), so \(\cos(980^\circ)=\cos(260^\circ)\approx-0.1736\). 2. \(-440^\circ+2\cdot 360^\circ=280^\circ\), so \(\sin(-440^\circ)=\sin(280^\circ)\approx-0.9848\). 3. \(-1234^\circ+4\cdot 360^\circ=206^\circ\), so \(\cos(-1234^\circ)=\cos(206^\circ)\approx-0.8988\). 4. \(1850^\circ-5\cdot 360^\circ=50^\circ\), so \(\sin(1850^\circ)=\sin(50^\circ)\approx0.7660\).

1. \(\alpha=260^\circ\); \(\cos(980^\circ)\approx-0.1736\) 2. \(\alpha=280^\circ\); \(\sin(-440^\circ)\approx-0.9848\) 3. \(\alpha=206^\circ\); \(\cos(-1234^\circ)\approx-0.8988\) 4. \(\alpha=50^\circ\); \(\sin(1850^\circ)\approx0.7660\)

- Add or subtract multiples of \(360^\circ\) to place each angle in the standard interval. - Use the quadrant of \(\beta\) to determine the sign. - Use a reference angle of \(30^\circ\), \(45^\circ\), or \(60^\circ\) to find the exact value.

1. \(405^\circ-360^\circ=45^\circ\), so \(\sin(405^\circ)=\sin(45^\circ)=\frac{\sqrt{2}}{2}\). 2. \(-300^\circ+360^\circ=60^\circ\), so \(\cos(-300^\circ)=\cos(60^\circ)=\frac{1}{2}\). 3. \(1020^\circ-2\cdot 360^\circ=300^\circ\), so \(\sin(1020^\circ)=\sin(300^\circ)=-\frac{\sqrt{3}}{2}\). 4. \(-120^\circ+360^\circ=240^\circ\), so \(\cos(-120^\circ)=\cos(240^\circ)=-\frac{1}{2}\).

a) \(\beta=45^\circ\); \(\frac{\sqrt{2}}{2}\) b) \(\beta=60^\circ\); \(\frac{1}{2}\) c) \(\beta=300^\circ\); \(-\frac{\sqrt{3}}{2}\) d) \(\beta=240^\circ\); \(-\frac{1}{2}\)

- Check that your calculator display indicates radian mode. - Use the fifth decimal digit to decide how to round the fourth decimal digit. - Enter a negative argument with its negative sign included.

1. Set the calculator to radian mode. 2. Evaluate and round each expression: a) \(\sin(0.85)\approx0.7513\) b) \(\cos(2.12)\approx-0.5220\) c) \(\sin(-0.5)\approx-0.4794\) d) \(\cos(7.5)\approx0.3466\)

a) \(0.7513\) b) \(-0.5220\) c) \(-0.4794\) d) \(0.3466\)

- Substitute each given value directly into the function rule. - You can enter the entire expression \(\sin(x)+\cos(x)\) in one calculator line. - Include the negative sign when entering the input in part c).

1. Substitute each input into \(f(x)=\sin(x)+\cos(x)\) and evaluate in radian mode. 2. For a), \(f(1.4)=\sin(1.4)+\cos(1.4)\approx1.1554169\), so \(f(1.4)\approx1.155\). 3. For b), \(f(3.5)=\sin(3.5)+\cos(3.5)\approx-1.2872399\), so \(f(3.5)\approx-1.287\). 4. For c), \(f(-2.1)=\sin(-2.1)+\cos(-2.1)\approx-1.3680555\), so \(f(-2.1)\approx-1.368\).

a) \(f(1.4)\approx1.155\) b) \(f(3.5)\approx-1.287\) c) \(f(-2.1)\approx-1.368\)

- Make sure your calculator is in radian mode. - A number closer to \(0\) has a smaller absolute value. - Use the fifth decimal digit to round to four decimal places.

1. Evaluate each expression in radian mode: a) \(\sin(4.5)\approx-0.9775\) b) \(\cos(-2.1)\approx-0.5048\) c) \(\sin(-0.75)\approx-0.6816\) d) \(\cos(10.2)\approx-0.7143\) 2. Compare absolute values: \(0.5048<0.6816<0.7143<0.9775\). Therefore, \(\cos(-2.1)\) is closest to \(0\).

a) \(-0.9775\) b) \(-0.5048\) c) \(-0.6816\) d) \(-0.7143\) The value \(\cos(-2.1)\) is closest to \(0\).

- Substitute the given values before evaluating the trigonometric functions. - Pay close attention to the negative sign in part c). - Keep full calculator precision until the final rounding step.

1. For a), \(\sin(1.5)\cdot\cos(-2.8)\approx-0.9398621\), so the result is \(-0.9399\). 2. For b), \(\cos(1.5)+\sin(-2.8)\approx-0.2642509\), so the result is \(-0.2643\). 3. For c), \(x+y=1.5+(-2.8)=-1.3\), and \(\sin(-1.3)\approx-0.9635582\), so the result is \(-0.9636\).

a) \(-0.9399\) b) \(-0.2643\) c) \(-0.9636\)

- Sine and cosine repeat after a rotation of \(2\pi\). - Add \(2\pi\) to a negative angle to obtain a positive coterminal angle. - Express \(2\pi\) with the same denominator as the given angle. - Check that each result lies in the required interval.

1. Sine and cosine have period \(2\pi\), so add or subtract multiples of \(2\pi\). 2. \(\frac{17\pi}{4}-4\pi=\frac{\pi}{4}\), so \(\sin\left(\frac{17\pi}{4}\right)=\sin\left(\frac{\pi}{4}\right)\). 3. \(-\frac{2\pi}{3}+2\pi=\frac{4\pi}{3}\), so \(\cos\left(-\frac{2\pi}{3}\right)=\cos\left(\frac{4\pi}{3}\right)\). 4. \(-\frac{7\pi}{6}+2\pi=\frac{5\pi}{6}\), so \(\sin\left(-\frac{7\pi}{6}\right)=\sin\left(\frac{5\pi}{6}\right)\). 5. \(\frac{19\pi}{6}-2\pi=\frac{7\pi}{6}\), so \(\cos\left(\frac{19\pi}{6}\right)=\cos\left(\frac{7\pi}{6}\right)\).

a) \(\sin\left(\frac{\pi}{4}\right)\) b) \(\cos\left(\frac{4\pi}{3}\right)\) c) \(\sin\left(\frac{5\pi}{6}\right)\) d) \(\cos\left(\frac{7\pi}{6}\right)\)

- Which coordinate of a point on the unit circle represents cosine, and which represents sine? - Compare \(35^\circ\) with the benchmark angles \(30^\circ\) and \(45^\circ\). - In Quadrant I, both coordinates are positive.

1. The angle \(35^\circ\) is in Quadrant I, so both coordinates are positive. It lies between the benchmark angles \(30^\circ\) and \(45^\circ\), so a reasonable estimate is \(P=(0.8, 0.6)\). 2. A calculator gives \(\cos(35^\circ) \approx 0.819\) and \(\sin(35^\circ) \approx 0.574\). 3. On the unit circle, the x-coordinate equals \(\cos(\alpha)\), and the y-coordinate equals \(\sin(\alpha)\).

a) Approximately \(P=(0.8, 0.6)\) b) \(\cos(35^\circ) \approx 0.819\) and \(\sin(35^\circ) \approx 0.574\). The x-coordinate is the cosine value, and the y-coordinate is the sine value.

- Imagine how a point moves on the unit circle as the angle increases. - What happens to the point’s height from \(0^\circ\) to \(90^\circ\)? - What happens to its horizontal coordinate? - When are the horizontal and vertical coordinates equal?

1. In Quadrant I, sine is the y-coordinate, which increases as the angle increases. Since \(75^\circ > 20^\circ\), \(\sin(75^\circ) > \sin(20^\circ)\). 2. In Quadrant I, cosine is the x-coordinate, which decreases as the angle increases. Since \(10^\circ < 80^\circ\), \(\cos(10^\circ) > \cos(80^\circ)\). 3. Sine and cosine are equal when the x- and y-coordinates are equal. In Quadrant I, this occurs on the line \(y=x\), at \(\alpha = 45^\circ\).

a) \(\sin(75^\circ)\) b) \(\cos(10^\circ)\) c) \(45^\circ\)

- What are the coordinates of a unit-circle point in terms of its angle? - How do coordinates change when a point is reflected across the y-axis? - Which quadrant contains a point with a negative x-coordinate and a positive y-coordinate?

1. A unit-circle point at angle \(\alpha\) has coordinates \((\cos(\alpha), \sin(\alpha))\). Thus, \(P=\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\). 2. Point \(Q=\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\) is the reflection of \(P\) across the y-axis, so it lies in Quadrant II. Therefore, \(\beta=180^\circ-60^\circ=120^\circ\). 3. The coordinates of \(Q\) give \(\cos(120^\circ)=-\frac{1}{2}\) and \(\sin(120^\circ)=\frac{\sqrt{3}}{2}\).

a) \(P\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\) b) \(\beta=120^\circ\) c) \(\sin(120^\circ)=\frac{\sqrt{3}}{2}\) and \(\cos(120^\circ)=-\frac{1}{2}\)

- Reduce angles by multiples of \(360^\circ\). - Recall the even and odd symmetries of cosine and sine. - Use the quadrant to check the sign. - Keep full precision until rounding.

1. Reduce \(510^\circ\) to \(150^\circ\): \(\sin(510^\circ)=\sin(150^\circ)=0.500\). 2. Since cosine is even, \(\cos(-130.4^\circ)=\cos(130.4^\circ)\approx-0.648\). 3. Reduce \(1000^\circ\) by \(720^\circ\): \(\sin(1000^\circ)=\sin(280^\circ)\approx-0.985\). 4. Reduce \(825^\circ\) by \(720^\circ\): \(\cos(825^\circ)=\cos(105^\circ)\approx-0.259\).

a) \(0.500\) b) \(-0.648\) c) \(-0.985\) d) \(-0.259\)

- Use degree mode on your calculator. - Locate \(310^\circ\) by dividing the unit circle into four quadrants. - Negative angles are measured clockwise. Find a positive coterminal angle for \(-215^\circ\).

1. For \(\gamma\), \(\sin(310^\circ)\approx-0.766\) and \(\cos(310^\circ)\approx0.643\). 2. The angle \(310^\circ\) lies in Quadrant IV, where x-coordinates are positive and y-coordinates are negative. Therefore, cosine is positive and sine is negative. 3. For \(\delta\), \(\sin(-215^\circ)\approx0.574\) and \(\cos(-215^\circ)\approx-0.819\).

a) \(\sin(310^\circ)\approx-0.766\); \(\cos(310^\circ)\approx0.643\). The angle is in Quadrant IV, so sine is negative and cosine is positive. b) \(\sin(-215^\circ)\approx0.574\); \(\cos(-215^\circ)\approx-0.819\)

- Add or subtract full rotations to reduce each angle. - Cosine is the x-coordinate on the unit circle. - Reflecting across the x-axis preserves the x-coordinate.

1. For \(480^\circ\), subtract one full rotation: \(480^\circ=120^\circ+1\cdot 360^\circ\). 2. The angle reflected across the x-axis is \(360^\circ-120^\circ=240^\circ\), and it has the same cosine. 3. For \(-150^\circ\), add one full rotation: \(-150^\circ=210^\circ-1\cdot 360^\circ\). 4. The reflected angle is \(360^\circ-210^\circ=150^\circ\), and it has the same cosine.

a) \(\alpha_1=120^\circ+1\cdot 360^\circ\); \(\alpha_2=210^\circ-1\cdot 360^\circ\) b) For \(\alpha_1\): \(240^\circ\); for \(\alpha_2\): \(150^\circ\)

- Add or subtract multiples of \(360^\circ\). - Coterminal angles have the same sine and cosine values. - Check the required inequalities for \(\beta_1\) and \(\beta_2\).

1. Subtract one full rotation: \(620^\circ-360^\circ=260^\circ\), so \(\alpha_0=260^\circ\). 2. A negative coterminal angle is \(260^\circ-360^\circ=-100^\circ\). 3. An angle greater than \(720^\circ\) is \(260^\circ+720^\circ=980^\circ\). 4. Both angles differ from \(260^\circ\) by integer multiples of \(360^\circ\), so they have the same sine value.

a) \(\alpha_0=260^\circ\) b) For example, \(\beta_1=-100^\circ\) and \(\beta_2=980^\circ\)

- Reduce large multiples of \(\pi\) by multiples of \(2\pi\). - Evaluate each trigonometric term separately. - Recall exact values at multiples of \(\frac{\pi}{2}\). - Pay attention to subtraction of a negative value.

1. Add \(6\pi\) to \(-\frac{13\pi}{2}\): \(\sin\left(-\frac{13\pi}{2}\right)=\sin\left(-\frac{\pi}{2}\right)=-1\). 2. Since \(2025\pi=1012\cdot 2\pi+\pi\), \(\cos(2025\pi)=\cos(\pi)=-1\). 3. \(\sin(4\pi)=0\) and \(\cos(-3\pi)=\cos(\pi)=-1\), so the sum is \(-1\). 4. \(\cos\left(-\frac{9\pi}{2}\right)=\cos\left(-\frac{\pi}{2}\right)=0\), while \(\sin\left(-\frac{\pi}{2}\right)=-1\). The difference is \(0-(-1)=1\).

a) \(-1\) b) \(-1\) c) \(-1\) d) \(1\)

- Use radian mode. - Reduce angles by multiples of \(2\pi\). - Use even and odd symmetry for negative angles. - Keep full precision until the final rounding step.

1. Reduce \(2.8\pi\) by \(2\pi\): \(\sin(2.8\pi)=\sin(0.8\pi)\approx0.588\). 2. A calculator in radian mode gives \(\cos(10)\approx-0.839\). 3. A calculator in radian mode gives \(\sin(-4.5)\approx0.978\). 4. Add \(2\pi\) to \(-\frac{13\pi}{6}\): \(\cos\left(-\frac{13\pi}{6}\right)=\cos\left(-\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}\approx0.866\).

a) \(0.588\) b) \(-0.839\) c) \(0.978\) d) \(0.866\)

- Compare each angle with multiples of \(\frac{\pi}{2}\). - Negative angles are measured clockwise. - Reduce angles greater than \(2\pi\). - Use the quadrant to check the sign.

1. Since \(\pi < 3.5 < \frac{3\pi}{2}\), the angle is in Quadrant III. Thus \(\cos(3.5)\approx-0.936\). 2. The angle \(-2\) is coterminal with \(2\pi-2\), which lies in Quadrant III. Thus \(\sin(-2)\approx-0.909\). 3. Reduce \(7.2\pi\) by \(6\pi\) to \(1.2\pi\), which lies in Quadrant III. Thus \(\cos(7.2\pi)\approx-0.809\). 4. The angle \(-\frac{4\pi}{9}\) lies between \(-\frac{\pi}{2}\) and \(0\), so its terminal side is in Quadrant IV. Thus \(\sin\left(-\frac{4\pi}{9}\right)\approx-0.985\).

a) \(-0.936\); Quadrant III b) \(-0.909\); Quadrant III c) \(-0.809\); Quadrant III d) \(-0.985\); Quadrant IV

- Reduce each angle to \([0, 2\pi)\). - Sine is the y-coordinate on the unit circle. - Use \(\sin(-x)=-\sin(x)\). - Compare angles with reference angle \(\frac{\pi}{6}\).

1. Use periodicity and unit-circle symmetry. 2. The expressions \(\sin\left(\frac{\pi}{6}\right)\), \(\sin\left(\frac{5\pi}{6}\right)\), \(\sin\left(\frac{13\pi}{6}\right)\), and \(\sin\left(-\frac{7\pi}{6}\right)\) all equal \(\frac{1}{2}\). 3. The expressions \(\sin\left(-\frac{\pi}{6}\right)\), \(\sin\left(\frac{7\pi}{6}\right)\), \(\sin\left(\frac{11\pi}{6}\right)\), and \(\sin\left(-\frac{5\pi}{6}\right)\) all equal \(-\frac{1}{2}\).

Value \(\frac{1}{2}\): \(\sin\left(\frac{\pi}{6}\right)\), \(\sin\left(\frac{5\pi}{6}\right)\), \(\sin\left(\frac{13\pi}{6}\right)\), \(\sin\left(-\frac{7\pi}{6}\right)\) Value \(-\frac{1}{2}\): \(\sin\left(-\frac{\pi}{6}\right)\), \(\sin\left(\frac{7\pi}{6}\right)\), \(\sin\left(\frac{11\pi}{6}\right)\), \(\sin\left(-\frac{5\pi}{6}\right)\)

- Reduce each angle to one rotation. - Identify the quadrant of the terminal side. - Use reference angles and quadrant signs. - Count full rotations for very large angles.

1. The angle \(220^\circ\) is in Quadrant III with reference angle \(40^\circ\). Thus \(\sin(220^\circ)\approx-0.64\) and \(\cos(220^\circ)\approx-0.77\). 2. The angle \(-30^\circ\) is coterminal with \(330^\circ\), in Quadrant IV. Thus \(\sin(-30^\circ)=-0.50\) and \(\cos(-30^\circ)\approx0.87\). 3. Reduce \(400^\circ\) to \(40^\circ\). Thus \(\sin(400^\circ)\approx0.64\) and \(\cos(400^\circ)\approx0.77\). 4. Reduce \(900^\circ\) to \(180^\circ\). Thus \(\sin(900^\circ)=0.00\) and \(\cos(900^\circ)=-1.00\).

a) \(\sin(220^\circ)\approx-0.64\); \(\cos(220^\circ)\approx-0.77\) b) \(\sin(-30^\circ)=-0.50\); \(\cos(-30^\circ)\approx0.87\) c) \(\sin(400^\circ)\approx0.64\); \(\cos(400^\circ)\approx0.77\) d) \(\sin(900^\circ)=0.00\); \(\cos(900^\circ)=-1.00\)

- Distinguish between the unitless angle \(2\) and the angle \(2^\circ\). - Which calculator mode is required for each term? - Evaluate the two sine values separately, but keep full precision until the final addition.

1. Interpret the unitless angle in radians: \(\sin(2) \approx 0.9093\). 2. Interpret the angle with the degree symbol in degrees: \(\sin(2^\circ) \approx 0.0349\). 3. Add using full calculator precision, then round: \(T = \sin(2) + \sin(2^\circ) \approx 0.9442\).

\(\sin(2) \approx 0.9093\) \(\sin(2^\circ) \approx 0.0349\) \(T \approx 0.9442\)

- Distinguish degree measures from radian measures. - Use the correct calculator mode for each expression. - Compare \(4\) with \(\pi\) and \(\frac{3\pi}{2}\). - Use quadrant signs to check the coordinates.

1. Evaluate the four expressions, using degrees for (1) and (2) and radians for (3) and (4): (1) \(\cos(110^\circ)\approx-0.342\) (2) \(\sin(260^\circ)\approx-0.985\) (3) \(\cos(4)\approx-0.654\) (4) \(\sin(-0.8)\approx-0.717\) 2. Since \(\pi < 4 < \frac{3\pi}{2}\), an angle of \(4\) radians has its terminal side in Quadrant III. Therefore, the point \(P\) lies in Quadrant III.

a) (1) \(-0.342\); (2) \(-0.985\); (3) \(-0.654\); (4) \(-0.717\) b) Quadrant III, because \(\pi < 4 < \frac{3\pi}{2}\)

- A full rotation is \(360^\circ\). - Write a formula for all angles coterminal with \(225^\circ\). - Test both positive and negative integer multiples of a full rotation. - Keep only angles in the given interval, and exclude the original angle.

1. Coterminal angles have the form \(\beta=225^\circ+360^\circ k\), where \(k\) is an integer. 2. Test integer values of \(k\) that keep \(\beta\) in the given interval. 3. For \(k=-3, -2, -1, 1\), the angles are \(-855^\circ\), \(-495^\circ\), \(-135^\circ\), and \(585^\circ\). The values for \(k=-4\) and \(k=2\) lie outside the interval, and \(k=0\) gives the original angle.

\(\beta \in \{-855^\circ, -495^\circ, -135^\circ, 585^\circ\}\)

- Track how the x- and y-coordinates change as a point moves from \(0^\circ\) to \(90^\circ\). - Use \(\sin(180^\circ-\theta)=\sin(\theta)\) and \(\cos(180^\circ-\theta)=-\cos(\theta)\). - Recall the exact sine values for \(30^\circ\) and \(60^\circ\). - Use signs first when ordering values.

1. In Quadrant I, cosine decreases from \(1\) to \(0\), while sine increases from \(0\) to \(1\). They are equal at \(45^\circ\), so \(\cos(\alpha)>\sin(\alpha)\) for \(0^\circ \le \alpha <45^\circ\). 2. The value \(\cos(160^\circ)\) is negative. Also, \(\sin(160^\circ)=\sin(20^\circ)\), and \(\cos(20^\circ)\) is positive and larger than \(\sin(20^\circ)\). Thus, \(\cos(160^\circ)<\sin(20^\circ)=\sin(160^\circ)<\cos(20^\circ)\). 3. \(\sin(30^\circ)=\frac{1}{2}\), so twice this value is \(1\). But \(\sin(60^\circ)=\frac{\sqrt{3}}{2}\ne1\). Therefore, sine is not proportional to the angle.

a) \(0^\circ \le \alpha <45^\circ\) b) \(\cos(160^\circ)<\sin(20^\circ)=\sin(160^\circ)<\cos(20^\circ)\) c) \(2\sin(30^\circ)=1\), but \(\sin(60^\circ)=\frac{\sqrt{3}}{2}\), so the claim is false.