Finding the Six Trigonometric Ratios for a Right Triangle

Q1. Find the 6 trigonometric ratios of $\theta$ for the triangle below:

Background

Topic: Right Triangle Trigonometry

This question tests your understanding of how to find the six basic trigonometric ratios (sine, cosine, tangent, cosecant, secant, cotangent) for a given angle in a right triangle.

Key Terms and Formulas

• Opposite side: The side opposite the angle $\theta$.

• Adjacent side: The side next to the angle $\theta$ (but not the hypotenuse).

• Hypotenuse: The longest side of the right triangle (opposite the right angle).

The six trigonometric ratios are:

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

• $\csc \theta = \frac{\text{hypotenuse}}{\text{opposite}}$

• $\sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}}$

• $\cot \theta = \frac{\text{adjacent}}{\text{opposite}}$

Step-by-Step Guidance

1. Identify the sides of the triangle relative to $\theta$. The side labeled 2 is adjacent to $\theta$, the side labeled 3 is the hypotenuse, and the side opposite $\theta$ is currently unknown.

2. Use the Pythagorean Theorem to find the length of the missing side (opposite to $\theta$): $a^2 + b^2 = c^2$ Here, $a$ is the unknown side, $b = 2$, and $c = 3$.

3. Set up the equation: $a^2 + 2^2 = 3^2$ and solve for $a$.

4. Once you have all three side lengths, write the six trigonometric ratios using the definitions above.

5. Simplify each ratio as much as possible, but do not calculate the final decimal values yet.

Try solving on your own before revealing the answer!

Final Answer:

The missing side is $\sqrt{5}$, so the six trigonometric ratios are:

• $\sin \theta = \frac{\sqrt{5}}{3}$

• $\cos \theta = \frac{2}{3}$

• $\tan \theta = \frac{\sqrt{5}}{2}$

• $\csc \theta = \frac{3}{\sqrt{5}}$

• $\sec \theta = \frac{3}{2}$

• $\cot \theta = \frac{2}{\sqrt{5}}$

These ratios are based on the side lengths relative to $\theta$ in the triangle.