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Values of trigonometric functions

Trigonometric ratios & functions

Japanese school year: Math II

What you learn

Learn how to evaluate sine, cosine, and tangent for general angles using the unit circle. These values are fundamental for describing circular motions, periodic phenomena, and wave behaviors. Prior familiarity with right-triangle trigonometry and angle definitions provides an excellent basis for mastering this topic.

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Key points

This formula calculates tan⁡θ\tan\theta by dividing sin⁡θ\sin\theta by cos⁡θ\cos\theta. Here, θ\theta is the angle, and it is used to express tangent in terms of sine and cosine.

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

For the same angle θ\theta, the sum of sin⁡2θ\sin^2\theta and cos⁡2θ\cos^2\theta is always 11. It is used to find cosine from sine, or vice versa.

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

This formula connects tan⁡θ\tan\theta and cos⁡θ\cos\theta for an angle θ\theta. It is used to calculate cos⁡θ\cos\theta directly from tan⁡θ\tan\theta.

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

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