Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 5 Trigonometric Functions 5.3 Part 2 Trigonometric Functions of Any Angle Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Objectives: Use the definitions of trigonometric functions of any angle. Use the signs of the trigonometric functions. Find reference angles. Use reference angles to evaluate trigonometric functions.

The Signs of the Trigonometric Functions

All Star Trig Class All Star Trig Class Use the phrase “All Star Trig Class” to remember the signs of the trig functions in different quadrants. All Star All functions are positive Sin is positive Trig Class Tan is positive Cos is positive

Example: Finding the Quadrant in Which an Angle Lies If name the quadrant in which the angle lies. lies in Quadrant III.

Your Turn #1: Finding the Quadrant in Which an Angle Lies If sin 𝜃<0 𝑎𝑛𝑑 tan 𝜃<0 name the quadrant in which the angle θ lies.

Your Turn #2: Finding the Quadrant in Which an Angle Lies If cos 𝜃 <0 𝑎𝑛𝑑 tan 𝜃<0 name the quadrant in which the angle θ lies.

Your Turn #3: Finding the Quadrant in Which an Angle Lies If csc 𝜃 >0 𝑎𝑛𝑑 tan 𝜃<0 name the quadrant in which the angle θ lies.

Your Turn #4: Finding the Quadrant in Which an Angle Lies If sec 𝜃 >0 𝑎𝑛𝑑 cot 𝜃 >0 name the quadrant in which the angle θ lies.

Example: Evaluating Trigonometric Functions Using the Example: Evaluating Trigonometric Functions Using the Quadrant in Which an Angle Lies Given tan 𝜃 =− 1 3 and cos 𝜃 <0, find sin θ and sec θ. Because both the tangent and the cosine are negative, θ lies in Quadrant II, sin θ + and sec θ −.

Your Turn: Evaluating Trigonometric Functions Using the Your Turn: Evaluating Trigonometric Functions Using the Quadrant in Which an Angle Lies Given sin 𝜃=− 2 5 𝑎𝑛𝑑 tan 𝜃>0 find cos 𝜃 𝑎𝑛𝑑 cot 𝜃.

Reference Angles The angles whose terminal sides fall in quadrants II, III, and IV will have values of sine, cosine and other trig functions which are identical (except for sign) to the values of angles in quadrant I. The acute angle which produces the same values is called the reference angle.

Reference Angles The reference angle is the angle between the terminal side and the nearest arm of the x-axis. The reference angle is the angle, with vertex at the origin, in the right triangle created by dropping a perpendicular from the point on the unit circle to the x-axis.

Definition of a Reference Angle A reference angle for an angle  is the positive acute angle made by the terminal side of angle  and the x-axis. (Shown below in red)

Quadrant II For an angle, , in quadrant II, the reference angle is    or 180° - θ. In quadrant II, sin() is positive cos() is negative tan() is negative Original angle Reference angle

Quadrant III For an angle, , in quadrant III, the reference angle is  -  or θ - 180°. In quadrant III, sin() is negative cos() is negative tan() is positive Original angle Reference angle

Quadrant IV For an angle, , in quadrant IV, the reference angle is 2   or 360° - θ. In quadrant IV, sin() is negative cos() is positive tan() is negative Reference angle Original angle

Procedure For Finding Reference Angles Quadrant II: ref. angle =    or 180° - θ Quadrant III: ref. angle is  -  or θ - 180°. Quadrant IV: ref. angle is 2   or 360° - θ.

Example #1: Find the reference angle for each angle. 218 Positive acute angle made by the terminal side of the angle and the x-axis is: 218  180 = 38 1387 First find coterminal angle between 0o and 360o Divide 1387 by 360 to get a quotient of about 3.9. Begin by subtracting 360 three times. 1387 – 3(360) = 307 The reference angle for 307 (in quadrant IV) is: 360 – 307  = 53

Example #2: Finding Reference Angles Find the reference angle, θ’ for each of the following angles: θ = 235° Solution: θ is in Quadrant III, therefore θ’ = θ - 180°, so θ’ = 235° - 180° = 55° θ = -4π/3 Solution: θ is negative, so find a positive coterminal < 2π; -4π/3 + 2π = -4π/3 + 6π/3 = 2π/3. 2π/3 is in Quadrant II, therefore θ’ = π – θ, so θ’ = π - 2π/3 = π/3

Your Turn #1: Finding Reference Angles Find the reference angle, for each of the following angles: a. b. c. d.

Your Turn #2: Finding Reference Angles Find the reference angle for each of the following angles: a. b. c.

Comparison of Trigonometric Functions of Angles vs Functions of Reference Angles Each angle below has the same reference angle Choosing the same “r” for a point on the terminal side of each (each circle same radius), you will notice from similar triangles that all “x” and “y” values are the same except for sign

Comparison of Trigonometric Functions of Angles vs Functions of Reference Angles Based on the observations on the previous slide: Trigonometric functions of any angle will be the same value as trigonometric functions of its reference angle, except for the sign of the answer The sign of the answer can be determined by quadrant of the angle Also, we previously learned that the trigonometric functions of coterminal angles always have equal values

Finding Trigonometric Function Values for Any Non-Acute Angle  Step 1 If  > 360, or if  < 0, then find a coterminal angle by adding or subtracting 360 as many times as needed to get an angle greater than 0 but less than 360. Step 2 Find the reference angle '. Step 3 Find the trigonometric function values for reference angle '. Step 4 Determine the correct signs for the values found in Step 3. (Hint: All Star Trig. Class.) This gives the values of the trigonometric functions for angle .

Example: Finding Exact Trigonometric Function Values of a Non-Acute Angle Find the exact values of the trigonometric functions for 210. (No Calculator!) Reference angle: 210 – 180 = 30 Remember side ratios for 30-60-90 triangle. Corresponding sides:

Example Continued Trig functions of any angle are equal to trig functions of its reference angle except that sign is determined from quadrant of angle 210o is in quadrant III where only tangent and cotangent are positive Based on these observations, the six trig functions of 210o are:

Your Turn: Finding Trig Function Values Using Reference Angles Find the exact value of: cos (240)

Example: Using Reference Angles to Evaluate Trigonometric Functions Use reference angles to find the exact value of Step 1 Find the reference angle, and Step 2 Use the quadrant in which lies to prefix the appropriate sign to the function value in step 1.

Your Turn #1: Using Reference Angles to Evaluate Trigonometric Functions Use reference angles to find the exact value of sin 300°.

Your Turn #2: Using Reference Angles to Evaluate Trigonometric Functions Use reference angles to find the exact value of