Guided Practice: Establishing Trigonometric Identities Using Right Triangles

Q2. Establish the identity using the right triangle shown.

Background

Topic: Trigonometric Identities and Right Triangles

This question is testing your ability to use the properties of right triangles and trigonometric ratios (sine, cosine, tangent) to establish a mathematical identity. The triangle has sides labeled 1 (opposite), x (adjacent), and u (hypotenuse), with angle $\theta$.

Key Terms and Formulas

• Sine: $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{u}$

• Cosine: $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{u}$

• Tangent: $\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{x}$

• Pythagorean Theorem: $u^2 = 1^2 + x^2$

Step-by-Step Guidance

1. Identify the sides of the triangle relative to angle $\theta$: the side labeled 1 is opposite, x is adjacent, and u is the hypotenuse.

2. Write the trigonometric ratios for $\theta$ using the triangle:

   • $\sin\theta = \frac{1}{u}$

   • $\cos\theta = \frac{x}{u}$

   • $\tan\theta = \frac{1}{x}$

3. Recall the Pythagorean Theorem for this triangle: $u^2 = 1^2 + x^2$.

4. Use the trigonometric ratios and the Pythagorean Theorem to set up an identity. For example, you might relate $\sin^2\theta + \cos^2\theta$ to 1.

5. Substitute the expressions for $\sin\theta$ and $\cos\theta$ in terms of $u$ and $x$ into the identity you are trying to establish.

Try solving on your own before revealing the answer!

Final Answer: $\sin^2\theta + \cos^2\theta = 1$

Substituting $\sin\theta = \frac{1}{u}$ and $\cos\theta = \frac{x}{u}$, we get:

$\left(\frac{1}{u}\right)^2 + \left(\frac{x}{u}\right)^2 = \frac{1 + x^2}{u^2} = 1$ (since $u^2 = 1 + x^2$).

This confirms the fundamental Pythagorean identity for trigonometric functions.