Trig Identity and Ratio

Statement

The trigonometric identities provide relationships between sine, cosine, and tangent:

• Pythagorean identity: \(\sin^2\theta + \cos^2\theta = 1\)
• Tangent ratio: \(\tan\theta = \frac{\sin\theta}{\cos\theta}\), provided \(\cos\theta \neq 0\).

Why it’s true

• In a right-angled triangle, \(\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}\).
• By Pythagoras’ theorem, \((\text{opposite})^2 + (\text{adjacent})^2 = (\text{hypotenuse})^2\). Dividing through by \((\text{hypotenuse})^2\) gives \(\sin^2\theta + \cos^2\theta = 1\).
• For tangent: \(\tan\theta = \frac{\text{opposite}}{\text{adjacent}}\). Substituting sine and cosine gives \(\tan\theta = \frac{\sin\theta}{\cos\theta}\).

Recipe (how to use it)

1. If you know \(\sin\theta\), you can find \(\cos\theta\) using \(\sin^2\theta + \cos^2\theta = 1\).
2. If you know \(\cos\theta\), you can find \(\sin\theta\).
3. If you know both sine and cosine, you can calculate tangent.

Spotting it

Look for questions where you are given one trigonometric ratio and asked to find another. If “prove” or “show that” appears, this identity is often the key.

Common pairings

• Trigonometric equations (e.g. solving \(\sin^2\theta = 1 - \cos^2\theta\)).
• Exact trig values for standard angles.

Mini examples

1. Given: \(\sin\theta = 3/5\). Find \(\cos\theta\). Solution: \(\cos^2\theta = 1 - (3/5)^2 = 1 - 9/25 = 16/25\). So \(\cos\theta = \pm 4/5\).
2. Given: \(\sin\theta = 0.6, \cos\theta = 0.8\). Then \(\tan\theta = 0.6/0.8 = 0.75\).

Pitfalls

• Forgetting that cosine can be negative depending on the quadrant.
• Using \(\tan = \sin \times \cos\) (wrong!) instead of \(\tan = \sin/\cos\).

Exam strategy

• Write down the identity first: \(\sin^2\theta + \cos^2\theta = 1\).
• Substitute the known value, then solve step by step.
• For tangent, always divide sine by cosine.

Summary

The two most important trig identities at GCSE and A-level are \(\sin^2\theta + \cos^2\theta = 1\) and \(\tan\theta = \sin\theta/\cos\theta\). They connect all three trig functions and are essential for solving trig equations, proving identities, and working with right-angled triangles.