1. Trigonometric Functions of Any Angle MATH 109 - Precalculus S. Rook

2. Overview Section 4.4 in the textbook: – Trigonometric functions of any angle – Reference angles – Trigonometric functions of real numbers 2

3. Trigonometric Functions of any Angle

4. Trigonometric Functions of Any Angle Given an angle θ in standard position and a point (x, y) on the terminal side of θ, then the six trigonometric functions of ANY ANGLE θ are can be defined in terms of x, y, and the length of the line connecting the origin and (x, y) denoted as r 4

5. Trigonometric Functions of any Angle (Continued) 5 FunctionAbbreviationDefinition The sine of θsin θ The cosine of θcos θ The tangent of θtan θ The cotangent of θcot θ The secant of θsec θ The cosecant of θcsc θ Where and x and y retain their signs from (x, y)

6. Trigonometric Functions of any Angle (Continued) 6 FunctionAbbreviationDefinition The sine of θsin θ The cosine of θcos θ The tangent of θtan θ The cotangent of θcot θ The secant of θsec θ The cosecant of θcsc θ Where and x and y retain their signs from (x, y)

7. Algebraic Signs of Trigonometric Functions The sign of the six trigonometric functions depends on which quadrant θ terminates in: r is the distance from the origin to (x, y) so it is ALWAYS positive – The signs of x and y depend on which quadrant (x, y) lies – Remember the shorthand notation involving “the element of” symbol: i.e. means theta is a standard angle which terminates in Q IV 7

8. Algebraic Signs of Trigonometric Functions (Continued) Functionsθ Є QIθ Є QIIθ Є QIIIθ Є QIV and++–– and+––+ and+–+– 8

9. Trigonometric Functions of any Angle (Example) Ex 1: Find the value of all six trigonometric functions if: a)(-1, 2) lies on the terminal side of θ b) (-7, -1) lies on the terminal side of θ 9

10. Trigonometric Functions of any Angle (Example) Ex 2: Given sec θ = - 3 ⁄ 2 where cos θ < 0, find the exact value of tan θ and csc θ 10

11. Reference Angles

12. An important definition is the reference angle – Allows us to calculate ANY angle θ using an equivalent positive acute angle We can now work in all four quadrants of the Cartesian Plane instead of just Quadrant I! Reference angle: denoted θ’, the positive acute angle that lies between the terminal side of θ and the x-axis θ MUST be in standard position 12

13. Reference Angles Examples – Quadrant I 13 Note that both θ and θ’ are 60°

14. Reference Angles Examples – Quadrant II 14

15. Reference Angles Examples – Quadrant III 15

16. Reference Angles Examples – Quadrant IV 16

17. Reference Angles Summary Depending in which quadrant θ terminates, we can formulate a general rule for finding reference angles: – For any positive angle θ, 0° ≤ θ ≤ 360°: If θ Є QI: θ’ = θ If θ Є QII: θ‘ = 180° – θ If θ Є QIII: θ‘ = θ – 180° If θ Є QIV: θ’ = 360° – θ 17

18. Reference Angles Summary (Continued) – If θ > 360°: Keep subtracting 360° from θ until 0° ≤ θ ≤ 360° Go back to the first step on the previous slide – If θ < 0°: Keep adding 360° to θ until 0° ≤ θ ≤ 360° Go back to the first step on the previous slide – If θ is in radians: Either replace 180° with π and 360° with 2π OR Convert θ to degrees 18

19. Reference Angles (Example) Ex 3: i) draw θ in standard position ii) draw θ’, the reference angle of θ: a)312°b) π ⁄ 8 c) 4π ⁄ 5 d) -127° e) 11π ⁄ 3 19

20. Trigonometric Functions of Real Numbers

21. Reference Angle Theorem Reference Angle Theorem: the value of a trigonometric function of an angle θ is EQUIVALENT to the VALUE of the trigonometric function of its reference angle – The ONLY thing that may be different is the sign Determine the sign based on the trigonometric function and which quadrant θ terminates in – The Reference Angle Theorem is the reason why we need to memorize the exact values of 30°, 45°, and 60° only in Quadrant I! 21

22. Evaluating a Trigonometric Function Exactly To evaluate a trigonometric function of θ: – Ensure that 0 < θ < 2π when using radians or 0° < θ < 360° when using degrees – Find θ’ the reference angle of θ – Evaluate the function using the EXACT values of the reference angle and the quadrant in which θ terminates Write the function in terms of sine or cosine if necessary 22

23. Evaluating a Trigonometric Function (Exactly) Ex 4: Give the exact value: a)sin 225°b) cos 750° c)tan 120°d) sec - 11π ⁄ 4 23

24. Summary After studying these slides, you should be able to: – Calculate the trigonometric function of ANY angle θ – State the reference angle of an angle θ in standard position – Evaluate a trigonometric function using reference angles and exact values Additional Practice – See the list of suggested problems for 4.4 Next lesson – Graphs of Sine & Cosine Functions (Section 4.5) 24