Fundamental Trigonometric Identities

Develop pythagorean relationships, quotients and valid cancellation.

Lesson 1 of 42 parts Practice Flashcards

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Pythagorean Relationships

Objective: Explain pythagorean relationships, apply the rule, and justify relevant restrictions.

The unit circle gives sin²θ+cos²θ=1. Dividing by cos²θ yields 1+tan²θ=sec²θ where cosine is nonzero; dividing by sin²θ yields 1+cot²θ=csc²θ where sine is nonzero. These are identities on their common domains. When recovering sine or cosine from a square, choose the sign using the angle’s quadrant rather than silently assuming positivity.

Formulas

$sin²θ+cos²θ=1$

Worked examples

Try itIf sin² θ=7/10, find cos² θ.Show answer

Step 1: Use sin² θ+cos² θ=1. Step 2: Subtract 7/10 from 1. Step 3: cos² θ=3/10.

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Pythagorean Relationships

Objective: Explain pythagorean relationships, apply the rule, and justify relevant restrictions.

Formulas

$sin²θ+cos²θ=1$

Worked examples

Try itIf sin² θ=7/10, find cos² θ.Show answer

Step 1: Use sin² θ+cos² θ=1. Step 2: Subtract 7/10 from 1. Step 3: cos² θ=3/10.

Quotients and Valid Cancellation

Objective: Explain quotients and valid cancellation, apply the rule, and justify relevant restrictions.

The identities tan θ=sin θ/cos θ and cot θ=cos θ/sin θ follow from coordinate ratios. Rewrite complicated expressions in sine and cosine when it reveals common factors. Cancellation is valid only for nonzero factors. An identity proof must preserve the original expression’s restrictions; a simpler final form can have a larger apparent domain than the starting expression.

Formulas

$tan θ=sin θ/cos θ, cos θ≠0$

Worked examples

Try itAt an angle with sin θ=7/10 and nonzero cos θ, evaluate tan θ·cos θ.Show answer

Step 1: Use tan θ=sin θ/cos θ. Step 2: Multiply by cos θ and cancel the nonzero denominator. Step 3: The result is sin θ=7/10.

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