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Algebra 262Updated Sep 2, 20262 pages

Master Trigonometry Identities

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Layla Guthrie@laylaguthrie_aoly
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Ready to master trigonometric identities? These powerful relationships between trig functions can make complex problems simple once you understand how they work. Let's break down the key identities and see how to apply them to verify and simplify expressions.
Trig Identities Practice – page 1

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Verifying and Simplifying Trigonometric Identities

Trigonometric identities are equations that are true for all values in their domain. The most fundamental identity to remember is sin²θ + cos²θ = 1, which serves as the foundation for many others.

When verifying identities, work with one side of the equation until it matches the other side. For example, in the identity cos⁡θtan⁡θ=sin⁡θ\cos\theta\tan\theta = \sin\theta, we can substitute tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta} to get cos⁡θ⋅sin⁡θcos⁡θ=sin⁡θ\cos\theta \cdot \frac{\sin\theta}{\cos\theta} = \sin\theta, which simplifies to sin⁡θ=sin⁡θ\sin\theta = \sin\theta.

To simplify expressions like tan⁡θcot⁡θ\tan\theta\cot\theta, remember that these are reciprocals, so tan⁡θcot⁡θ=sin⁡θcos⁡θ⋅cos⁡θsin⁡θ=1\tan\theta\cot\theta = \frac{\sin\theta}{\cos\theta} \cdot \frac{\cos\theta}{\sin\theta} = 1. Similarly, sin⁡θcsc⁡θ=1\sin\theta\csc\theta = 1 and cos⁡θsec⁡θ=1\cos\theta\sec\theta = 1 because csc and sec are reciprocals of sin and cos respectively.

Pro Tip: When simplifying complex expressions, try converting everything to sines and cosines first, then look for patterns like sin²θ + cos²θ = 1 or sec²θ - tan²θ = 1 to help simplify further.

Trig Identities Practice – page 2

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Applying Identities to Find Unknown Values

When you know one trigonometric function and the quadrant of the angle, you can find all other trig functions. This is super useful for solving problems without a calculator!

For example, if sin⁡θ=0.48\sin\theta = 0.48 and θ is in the second quadrant, we can find cos⁡θ\cos\theta using the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1. Substituting gives us (0.48)2+cos⁡2θ=1(0.48)^2 + \cos^2\theta = 1, so cos⁡2θ=1−0.2304=0.7696\cos^2\theta = 1 - 0.2304 = 0.7696. Since θ is in the second quadrant, cos⁡θ\cos\theta is negative, so cos⁡θ=−0.7696≈−0.877\cos\theta = -\sqrt{0.7696} ≈ -0.877. Then we can find tan⁡θ=sin⁡θcos⁡θ=0.48−0.877≈−0.547\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{0.48}{-0.877} ≈ -0.547.

The unit circle is an incredibly helpful tool for visualizing these relationships. It can help you verify identities like sin⁡(θ+π)=−sin⁡θ\sin(\theta + \pi) = -\sin\theta by showing how angles related by π radians appear on opposite sides of the circle.

Remember: In different quadrants, the signs of trig functions change! In the first quadrant, all trig functions are positive; in the second, only sin is positive; in the third, only tan is positive; and in the fourth, only cos is positive. The mnemonic "All Students Take Calculus" can help you remember this pattern.

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