Trigonometric Identities and Equations

Explore pythagorean identity, reciprocal and quotient rules, solutions on an interval.

Lesson 4 of 53 parts Practice Flashcards

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Pythagorean Identity

Objective: Explain pythagorean identity and use it to solve or interpret a problem.

sin²θ+cos²θ=1 follows from x²+y²=1 on the unit circle. Divide only when the divisor is nonzero to derive tangent identities. Division by zero would be invalid.

Worked examples

Try itTrigonometric Identities and Equations — practice 1.1: If sin²θ=4/7, find cos²θ.Show answer

Step 1: Use sin²θ+cos²θ=1. Step 2: Subtract 4/7 from 1. Step 3: cos²θ=3/7.

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Pythagorean Identity

Objective: Explain pythagorean identity and use it to solve or interpret a problem.

sin²θ+cos²θ=1 follows from x²+y²=1 on the unit circle. Divide only when the divisor is nonzero to derive tangent identities. Division by zero would be invalid.

Worked examples

Try itTrigonometric Identities and Equations — practice 1.1: If sin²θ=4/7, find cos²θ.Show answer

Step 1: Use sin²θ+cos²θ=1. Step 2: Subtract 4/7 from 1. Step 3: cos²θ=3/7.

Reciprocal and Quotient Rules

Objective: Explain reciprocal and quotient rules and use it to solve or interpret a problem.

Where defined, tanθ=sinθ/cosθ, secθ=1/cosθ, and cscθ=1/sinθ. State where denominators vanish. When cosθ=0.

Worked examples

Try itTrigonometric Identities and Equations — practice 2.1: Find tan(3π/4).Show answer

Step 1: Locate 3π/4 on the unit circle. Step 2: Divide its sine by its nonzero cosine. Step 3: The tangent value is -1.

Solutions on an Interval

Objective: Explain solutions on an interval and use it to solve or interpret a problem.

A periodic equation can have multiple solutions; use unit-circle positions and report only values inside the stated interval with endpoint rules. Without one, periodic equations have infinitely many shifted solutions.

Worked examples

Try itTrigonometric Identities and Equations — practice 3.1: How many solutions does sin x=0 have on 0<x<2π?Show answer

Step 1: Sine is zero at multiples of π. Step 2: Check which of 0, π, and 2π belong to 0<x<2π. Step 3: There are 1 included solutions.

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