Objectives: Students will learn how to find Cos, Sin & Tan using the special right triangles.

The 45-45-90 Special Right Triangle Vocabulary SOHCAHTOA The 30-60-90 Special Right Triangle The 45-45-90 Special Right Triangle The Unit Circle

The unit circle is used to understand sines and cosines of angles found in right triangles. The unit circle has a center at the origin (0,0) and a radius of 1-unit. Angles are measured starting from the positive x-axis in quadrant I and continue around the unit circle.

Special Right Triangle Special Right Triangle The 45-45-90 Special Right Triangle The 30-60-90 Special Right Triangle

Special Right Triangle Special Right Triangle State the: 45-45-90 Special Right Triangle B State the: 30-60-90 Special Right Triangle A

60 60 Find the sine, cosine, and tangent of 30°:

Find the sine, cosine, and tangent of 45°:

From the coordinates given, you know that x = 8 and y = –15. Evaluate Trigonometric Functions Given a Point The terminal side of  in standard position contains the point (8, –15). Find the exact values of the six trigonometric functions of . 8 From the coordinates given, you know that x = 8 and y = –15. –15 Use the Pythagorean Theorem to find r. Pythagorean Theorem Replace x with 8 and y with –15. Simplify. Now use x = 8, y = –15, and r = 17 to write the ratios. Example 1

Find the exact values of the six trigonometric functions of  if the terminal side of  contains the point (–3, 4). Example 1

What is a Reference angle? The reference angle is the smallest angle between the terminal side and the x-axis.

Find the reference angle of 330°. Find Reference Angles Find the reference angle of 330°. Answer: Because the terminal side of 330 lies in quadrant IV, the reference angle is 360 – 330 = 30. Example 3

Find the reference angle of 315°. Find Reference Angles Find the reference angle of 315°. Answer: Because the terminal side of 330 lies in quadrant IV, the reference angle is 360 – 315 = 45. Example 3

The values of the other degree trig functions of angle > 90 are the same as the trig values of the angle < 90, give or take a minus sign.

Trigonometric Functions and Reference Angles

Example 3: Using Reference Angles to Evaluate Trigonometric functions Use a reference angle to find the exact value of the sine, cosine, and tangent of 330°. Step 1 Find the measure of the reference angle. The reference angle measures 360-330 = 30°

Example 3 Continued Step 2 Find the sine, cosine, and tangent of the reference angle. Use sin θ = y. Use cos θ = x.

Example 3 Continued Step 3 Adjust the signs, if needed. In Quadrant IV, sin θ is negative. In Quadrant IV, cos θ is positive. In Quadrant IV, tan θ is negative.

A. Find the exact value of sin 135°. Use a Reference Angle to Find a Trigonometric Value A. Find the exact value of sin 135°. Because the terminal side of 135° lies in Quadrant II, the reference angle  ' is 180° – 135° or 45°. Answer: The sine function is positive in Quadrant II, so, sin 135° = sin 45° or Example 4

Evaluate the trigonometric function without using a calculator Evaluate the trigonometric function without using a calculator. Angles are given in degrees. cos 150° Find the reference angle for Quadrant II. Use the sides of a 30°−60°−90° triangle, then apply the sign of cosine in Quadrant II.

A. Find the exact value of sin 120°. B. C. D. Example 4

Evaluate the trigonometric function without using a calculator Evaluate the trigonometric function without using a calculator. Angles are given in degrees. tan(−60°) Find the reference angle for Quadrant IV. Use the sides of a 30°−60°−90° triangle, then apply the sign of cosine in Quadrant IV.

Use Trigonometric Functions RIDES The swing arms of the ride pictured below are 89 feet long and the height of the axis from which the arms swing is 99 feet. What is the total height of the ride at the peak of the arc? Example 5

coterminal angle: –200° + 360° = 160° Use Trigonometric Functions coterminal angle: –200° + 360° = 160° reference angle: 180° – 160° = 20° sin  Sine function  = 20° and r = 89 89 sin 20° = y Multiply each side by 89. 30.4 ≈ y Use a calculator to solve for y. Answer: Since y is approximately 30.4 feet, the total height of the ride at the peak is 30.4 + 99 or about 129.4 feet. Example 5

RIDES The swing arms of the ride pictured below are 68 feet long and the height of the axis from which the arms swing is 79 feet. What is the total height of the ride at the peak of the arc? A. 23.6 ft B. 79 ft C. 102.3 ft D. 110.8 ft Example 5