1.4 Trigonometric Functions of Any Angle Part 3

Finding Reference Angles Ex: Find the values of the trigonometric functions for 210º. Solution: The reference angle for 210º is 210º – 180º = 30º. A 30º-60º-90º right triangle is formed.

Finding Trigonometric Function Values Using Reference Angles Example Find the exact value of each expression. (a) cos(–240º) (b) tan 675º Solution a) –240º is coterminal with 120º. The reference angle is 180º – 120º = 60º. Since –240º lies in quadrant II, the cos(–240º) is negative. (b) tan 675º = tan 315º = –tan 45º = –1.

So what’s so great about reference angles? To find the value of the trig function of any non-acute angle, we just need to find the trig function of the reference angle. After, we determine whether it is positive or negative, depending upon the quadrant in which the angle lies. For example, In Quad 3, sin is negative 45° is the ref angle

Evaluating Trig Functions of Common Angles Summary: Evaluating Trig Functions of Common Angles To find the value of a trig function for any common angle  Determine the quadrant in which the angle lies. Determine the reference angle. Use one of the special triangles to determine the function value for the reference angle. Depending upon the quadrant in which  lies, use the appropriate sign (+ or –).

Trig Functions of Common Angles Using reference angles and the special reference triangles, we can find the exact values of the common angles. 30°-60° right triangle

Just a Reminder: Trig Functions of Common Angles θ (in degrees) θ (in radians) 30°-60° right triangle

Using Trigonometric Identities Let θ be an angle in Quadrant II such that sin θ= 1/3. Find (a) cos θ and (b) tan θ by using the trig identities. Solution: (a) Use the Pythagorean Identity: sin²θ + cos²θ = 1 (1/3)² + cos²θ = 1 cos²θ = 1 – 1/9 = 8/9 Because x is negative in Quadrant 2, cosine must be negative: cos θ = - √8 = - 2√2 √9 3 (b) Using the trig identity tan θ = sin θ/cos θ: tan θ = 1/3 = -1 = -√2 -2√2/3 2√2 4

Evaluating trig function of “uncommon” angles To find the value of the trig functions of angles that do NOT reference 30°, 45°, or 60°, and are not quadrantal, we will use the calculator. Round your answer to 4 decimal places, if necessary. Make sure the Mode setting is set to the correct form of the angle: Radian or Degree To find the trig functions of csc, sec, and cot, use the reciprocal identities.

Finding Trigonometric Function Values with a Calculator Example Approximate the value of each expression. (a) cos 49º 12 (b) csc 197.977º Solution Set the calculator in degree mode. (a) cos 49º 12 = cos 49.2 = .653420604 (b) csc 197.977º = 1/sin(197.977) = -3.240071221

Finding Angle Measure Example Using Inverse Trigonometric Functions to Find Angles Use a calculator to find an angle  in degrees that satisfies sin   .9677091705. Use a calculator to find an angle  in radians that satisfies tan   .25. Solution With the calculator in degree mode, an angle having a sine value of .9677091705 is 75.4º. Write this as sin-1 .9677091705  75.4º. With the calculator in radian mode, we find tan-1 .25  .2449786631.

End of Section1.4

Classwork: Worksheet – Exact Trig Values of Special Angles Section 1.4 Quest on Friday

Homework: