What are Trigonometric Identities?
Trigonometric identities are equations involving sine, cosine, tangent and related functions that hold true for every value of the angle. They let you simplify expressions, solve equations and switch between equivalent trig forms.
The core Pythagorean identity is sin²θ + cos²θ = 1, true for every angle θ. It follows directly from the Pythagorean theorem applied to the unit circle.
- •sin²θ + cos²θ = 1
- •1 + tan²θ = sec²θ
- •1 + cot²θ = csc²θ
- •sin(A+B) = sinA cosB + cosA sinB
- •cos(A+B) = cosA cosB − sinA sinB
- •tan(A+B) = (tanA+tanB)/(1−tanA tanB)
Try it: interactive calculator
Step-by-step worked examples
If sin θ = 0.6, find cos θ (θ in the first quadrant).
sin²θ + cos²θ = 1 0.6² + cos²θ = 1 0.36 + cos²θ = 1 cos²θ = 0.64 cos θ = 0.8
Simplify sin²θ + cos²θ + 5.
sin²θ + cos²θ = 1 (identity) Result = 1 + 5 = 6
If cos θ = 0.8, find tan θ using sin θ from the Pythagorean identity.
sin²θ = 1 − cos²θ = 1 − 0.64 = 0.36 sin θ = 0.6 tan θ = sin θ/cos θ = 0.6/0.8 = 0.75
Flashcards
Quick quiz
Q1.sin²θ + cos²θ equals:
Q2.1 + tan²θ equals:
Q3.If sin θ = 0.6, cos θ (Q1) is:
Q4.sin(2θ) equals:
The full card deck, worked steps and AI-tutor support for “What are Trigonometric Identities?” are in Notek — study by hand before your exam.
Common mistakes
Writing sin²θ as sin(θ²). — Correct: sin²θ means (sin θ)², squaring the ratio, not squaring the angle.
Assuming identities only work for special angles. — Correct: Trig identities hold for every value of θ, not just 30°/45°/60°.
Forgetting the sign of cos θ or sin θ outside quadrant I. — Correct: Use the quadrant to decide the ± sign when taking a square root.
Confusing sum identities with double-angle identities. — Correct: sin(A+B) needs two angles; sin(2θ) is the special case A=B=θ.
FAQ
What are trigonometric identities?
Equations relating trig functions (sin, cos, tan…) that are true for every angle value.
What is the trigonometric identities formula?
The core one is sin²θ + cos²θ = 1; others include 1+tan²θ=sec²θ and angle-sum formulas.
How do you calculate with trigonometric identities?
Substitute a known ratio into an identity and solve algebraically for the unknown one.
What are examples of trigonometric identities?
sin²θ+cos²θ=1, 1+tan²θ=sec²θ, and sin(2θ)=2sinθcosθ are classic examples.