Trigonometry • Topic 4 of 4

Simple Identities Involving sin, cos, and tan

What are Trigonometric Identities?

A Trigonometric Identity is an algebraic equation involving trigonometric functions that remains true for every single value substituted into the angle variable. Unlike a regular equation that you solve for a specific x value, an identity is a universal rule of balance.

There are two major sets of simple identities that form the foundation of triangle algebra:

  1. Quotient Identities: These identities show how the tangent ratio is built directly by dividing the sine ratio by the cosine ratio.
  • tan theta = sin theta / cos theta
  • cot theta = cos theta / sin theta
  1. Pythagorean Identities: These identities are derived by combining the Pythagoras Theorem (\(a^2 + b^2 = c^2\)) with our fundamental trigonometric fraction definitions. The most important foundational Pythagorean identity is:
  • sin² theta + cos² theta = 1

Think of this like an algebraic currency exchange. If you are working through a long mathematical expression and you have a mix of sine terms and cosine terms, you can use these identities to trade them out and rewrite everything in a single clean form.

Heights & Distances — Angle of Elevation/Depressionhd (distance)θ (elevation)TopEyeBaseProblem-Solving Templatetan θ = Opposite / Adjacent = h / d∴ h = d × tan θ (height from angle and distance)∴ d = h / tan θ (distance from angle and height)Use sin θ = h/hyp when hypotenuse is involvedAngle of depression: same magnitude as angle of elevation (alternate angles)Always draw a diagram first — label known and unknown quantities
Solved Examples
1
Worked Example

Solve a standard problem on Simple Identities Involving sin, cos, and tan.

Solution

Apply the formula/method shown in the concept section above.

Key Points

  • Understand the definition and properties of Simple Identities Involving sin, cos, and tan.
  • Study the worked examples and practice similar problems.
  • Always verify your answer using the original conditions.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.$\sin^2\theta+\cos^2\theta=$
Explanation: Fundamental identity.
Q2.$1+\tan^2\theta=$
Explanation: $\sec^2\theta$.
Q3.$1+\cot^2\theta=$
Explanation: $\operatorname{cosec}^2\theta$.
Q4.$\sec^2\theta-\tan^2\theta=$
Explanation: $1$.
Practice more MCQs → 📝 Assignment · 35 marks