Trigonometry Unit 5: Trigonometric Identities — Free Review Games.
This unit covers Pythagorean identities, sum and difference formulas, double-angle formulas and half-angle formulas — essential concepts for Trigonometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
Pick a mode. Play.
Answer questions as fast as you can. 2 minutes on the clock. Build streaks for bonus points!
Don't want to play?
Review the questions traditionally. Click to expand.
Questions loading...
Focus on understanding.
Focus on understanding core concepts before memorizing details. Use the game modes to test yourself repeatedly — spaced repetition is proven to boost long-term retention.
Related units
This unit covers Pythagorean identities, sum and difference formulas, double-angle formulas and half-angle formulas — essential concepts for Trigonometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Pythagorean Identities
Students must memorize the three Pythagorean identities and be able to derive the second and third from the first. On exams, these are used to simplify expressions and verify other identities by substituting equivalent forms.
Key Points
- sin²θ + cos²θ = 1 is the foundational identity — memorize it cold
- Dividing by cos²θ gives tan²θ + 1 = sec²θ; dividing by sin²θ gives 1 + cot²θ = csc²θ
- Use these to replace sin²θ with 1 − cos²θ (or vice versa) to simplify one side of an identity
- Never cross-multiply when verifying identities — only manipulate one side at a time
Simplify: (1 − cos²θ)(csc²θ)
Replace 1 − cos²θ with sin²θ using the first Pythagorean identity, giving sin²θ · csc²θ. Since cscθ = 1/sinθ, this becomes sin²θ · (1/sin²θ) = 1. The expression simplifies to 1.
2 Sum and Difference Formulas
Students must know the sum and difference formulas for sine and cosine and apply them to find exact values of non-standard angles. Exams frequently ask for exact values of angles like 75° or 15° that can be written as sums or differences of 30°, 45°, or 60°.
Key Points
- sin(A ± B) = sinA cosB ± cosA sinB (sign matches outside)
- cos(A ± B) = cosA cosB ∓ sinA sinB (sign flips — opposite outside)
- tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)
- To find an exact value, rewrite the angle as a sum/difference of two known reference angles
Find the exact value of sin(75°).
Rewrite 75° as 45° + 30° and apply the formula: sin(45° + 30°) = sin45°cos30° + cos45°sin30°. Substituting exact values: (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4.
3 Double-Angle Formulas
Students must know all three forms of the cosine double-angle formula and recognize when to use each. Exams test both finding exact values given a trig ratio and simplifying expressions using these formulas.
Key Points
- sin(2θ) = 2sinθ cosθ
- cos(2θ) has three equivalent forms: cos²θ − sin²θ, 2cos²θ − 1, or 1 − 2sin²θ — choose the one matching what is given
- tan(2θ) = 2tanθ / (1 − tan²θ)
- If given sinθ and cosθ, use sin(2θ) = 2sinθ cosθ directly without finding θ first
If sinθ = 3/5 and θ is in Quadrant I, find sin(2θ) and cos(2θ).
First find cosθ using sin²θ + cos²θ = 1: cosθ = 4/5 (positive in QI). Then sin(2θ) = 2(3/5)(4/5) = 24/25. For cos(2θ), use cos²θ − sin²θ = (16/25) − (9/25) = 7/25.
4 Half-Angle Formulas
Students must know the half-angle formulas for sine and cosine and determine the correct sign based on the quadrant of the half-angle (not the original angle). Exams use these to find exact values of angles like 22.5° or 157.5°.
Key Points
- sin(θ/2) = ±√((1 − cosθ)/2); cos(θ/2) = ±√((1 + cosθ)/2)
- The ± sign is determined by the quadrant where θ/2 lies, not where θ lies
- tan(θ/2) = sinθ/(1 + cosθ) = (1 − cosθ)/sinθ (these forms have no ± ambiguity)
- The argument inside the formula is the full angle θ, and you take half of it as the result
Find the exact value of cos(22.5°).
Recognize that 22.5° = 45°/2, so use the half-angle formula with θ = 45°: cos(22.5°) = √((1 + cos45°)/2) = √((1 + √2/2)/2) = √((2 + √2)/4) = √(2 + √2)/2. The sign is positive because 22.5° is in Quadrant I.
Questions, answered.
What is Trigonometric Identities?
Trigonometric Identities is Unit 5 of Trigonometry, covering Pythagorean identities, sum and difference formulas, double-angle formulas and half-angle formulas.
How to study for Trigonometry Unit 5?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 28+ review questions across 5 different game modes.