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Trigonometric ratios

Solving triangles

Japanese school year: Math I

What you learn

Understand definitions and identities of sine, cosine, and tangent based on right triangles. Building on triangle similarity and the Pythagorean theorem, learners calculate unknown side lengths from an angle and a known side. These ratios are foundational for surveying, physics, and vector decomposition.

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Key points

This fundamental identity connects sin⁡θ\sin\theta and cos⁡θ\cos\theta for any angle θ\theta. It is used to find one value when the other is known, since the sum of their squares is always 1.

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1

This formula defines tan⁡θ\tan\theta as the quotient of sin⁡θ\sin\theta divided by cos⁡θ\cos\theta. Use it to find tangent directly from known sine and cosine values.

tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}

This identity directly links tan⁡θ\tan\theta and cos⁡θ\cos\theta. It is used to compute cos⁡θ\cos\theta directly from tan⁡θ\tan\theta without having to calculate sin⁡θ\sin\theta first.

1+tan⁡2θ=1cos⁡2θ1 + \tan^2\theta = \frac{1}{\cos^2\theta}

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