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Using Fundamental Trigonometric Identities and the Unit Circle to Find Missing Values

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In this chapter, we explore how to utilize fundamental trigonometric identities and the unit circle to determine missing trigonometric values. By starting with known trigonometric functions, we can ascertain the appropriate quadrant for the angle θ and employ the Pythagorean theorem to find other side lengths of a triangle. This allows the calculation of all six trigonometric functions. We also cover various identity groups, including reciprocal, quotient, cofunction, Pythagorean, and even-odd identities. Practical exercises and hints from previous chapters are included for effective problem-solving.

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Using Fundamental Trigonometric Identities and the Unit Circle to Find Missing Values

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  1. Chapter 5.1 Using Fundamental Identities

  2. Using the unit circle to find remaining trigonometric values • When given a trig function, there are two values of the triangle’s sides that are inherently given to us: • When given two trig functions, it is possible to determine which quadrant/axis contains θ • Using these values and the Pythagorean Thm it is possible to find the remaining side • Using all three sides it is possible to find the remaining trig functions

  3. Using previous notes and the back cover of the book reference sheet for all the formulas to help you through some of these problems. • Hint: The ones you memorized in Chapter 4 

  4. The fundamental trigonometric identities come in several related groups: • Reciprocal Identities • Quotient Identities • Cofunction Identities • Pythagorean Identities • Even-Odd Identities

  5. Ex 1: Given a cscθand tan θUsing the ∆ to solve for the trig f(x) Quad 1 3 θ 4

  6. Ex 1: Given a cscθand tan θ 5 3 θ 4

  7. Ex 1: Given a cscθand tan θ 5 3 θ 4

  8. Ex 2: Given a cot θand cosθUsing the Unit Circle to Solve for the Trig f(x) Θ must be in Quad 1, Quad 4 or the positive x-axis

  9. Ex 2: Given a cot θand cosθ Θ must be in Quad 1, 4 or the positive x-axis sin θ = + cos θ = - tan θ = - sin θ = + cos θ = + tan θ = + Q2 (-, +) Q1 (+, +) Q3 (-, -) Q4 (+, -) sin θ = - cos θ = + tan θ = - sin θ = - cos θ = - tan θ = + Θ must be 0

  10. Ex 2: Given a cot θand cosθ • You can’t draw a triangle with θ = 0, • But you do have the unit circle memorized as • (1, 0) • which allows you to find the six trig functions.

  11. Proving Trig Functions are equal • On top of the previous notes, there are some equivalent trig functions

  12. Verifying Identities In order to verify an equation is an identity, you must follow these steps: • Start with the expression on one side of the equation • (Hint: pick the “harder” side.) • Manipulate that expression using known identities • (Hint: put everything into sin θand cosθ) • Stop when you reach the expression on the other side of the equation

  13. Ex 3: Verify the identity

  14. Ex 4: Verify the identity OOOh, there’s a common factor!

  15. Ex 5: Verifying the identity

  16. Ex 5: Verifying the identity

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