5.2 Circles and the Sine Ratio

Prerequisite Skills 

1. Use the sine, cosine, or tangent ratio to determine the length of the side indicated, to the nearest unit.

1. Find two coterminal angles for each given angle. Draw each set of coterminal angles on the same sex of axes in standard position.

a) 30°     b) 270°    c) -60°

2. Which pairs of angles are coterminal? Justify your answer.

a) 45° and 450°     b) -60° and 300°     c) 180° and 900°   d) 271° and -239°

3. An angle is in standard position and has terminal point P. For each set of coordinates for P, find the radius of the circle in exact form and the measure of the angle to the nearest tenth of a degree. Include a diagram illustrating each angle.

Plot the point, draw a right triangle using the terminal arm with point Pas the hypotenuse, and the x-axis as the base of the triangle. Complete the triangle. Apply the Pythagorcan theorem.

a) P(-3,-6)     b) P(4, - 7)

4. Find the coordinates of a terminal point, P(x, y), for each angle in standard position on the unit circle. Round to one decimal place. Draw a diagram to illustrate each angle.

Determine the sine and cosine of the given angle for a unit circle. Determine the values of X and y using the Pythagorean theorem.

a) 60°     b) 135°    c) 310°

5. Verify thata9 cos θ) + (sin θ)2 = 1 for each of the following angles.

a) 0 = 21°     b) 0 = 315°

6. Evaluate. Round to three decimal places. Explain why the sign of the result is either positive or negative.

a) tan 210°    b) sin 300°    c) cos 120°

7. Find the measure of angle 0 and find one coterminal angle with the same value of sin 9. Then find one non-coterminal angle that gives the same value of sin 0. Round your answers to the nearest degree.

a) sin θ = 0.8090    b) sin θ = -0.7071

8. Determine the measure of angle 0 in standard position, to the nearest degree.

a) sin θ = -0.6427 if θ is in the fourth quadrant

b) cos θ = -0.2588 if θ is in the third quadrant

c) tan θ = -1.6643 if θ is in the second quadrant

9. The radius of a carousel is 8 m. Starting at the entrance, a rider rotates around the circumference of the carousel. Find the distance from the rider to the entrance of the carousel at each of the following angles.

Draw a diagram.

a) θ = 30°     b) θ - 190°    c) θ = 310°

10. Given angle θ in standard position with terminal arm in the third quadrant and sin θ = -24/25 determine the value of cos θ + sin θ + tan θ.

11. A circle is drawn on graph paper with the origin at its centre. What is the area of the smaller sector bound by the radii reaching to (6.2) and (-2.6), to the nearest tenth?