To solve for the values of $\sin \theta$, $\cos \theta$, and $\tan \theta$ in this right triangle, let's analyze the sides in relation to $\theta$.

Given:

• Opposite side (to $\theta$): 8
• Adjacent side (to $\theta$): 7

To find the hypotenuse, $h$, we use the Pythagorean theorem: $h = \sqrt{7^2 + 8^2} = \sqrt{49 + 64} = \sqrt{113}$

Now, we can find each trigonometric ratio:

1. Sine $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$: $\sin \theta = \frac{8}{\sqrt{113}}$

2. Cosine $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$: $\cos \theta = \frac{7}{\sqrt{113}}$

3. Tangent $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$: $\tan \theta = \frac{8}{7}$

To provide the exact values in a simplified form, we could rationalize the denominators for $\sin \theta$ and $\cos \theta$:

• $\sin \theta = \frac{8}{\sqrt{113}} \times \frac{\sqrt{113}}{\sqrt{113}} = \frac{8 \sqrt{113}}{113}$
• $\cos \theta = \frac{7}{\sqrt{113}} \times \frac{\sqrt{113}}{\sqrt{113}} = \frac{7 \sqrt{113}}{113}$

Final Answers

• $\sin \theta = \frac{8 \sqrt{113}}{113}$
• $\cos \theta = \frac{7 \sqrt{113}}{113}$
• $\tan \theta = \frac{8}{7}$

Would you like a deeper explanation on any part of this solution or further examples?

Here are some related questions to explore further:

1. How would you find the values if only the hypotenuse and one leg were given?
2. What are the reciprocal trigonometric functions (cosecant, secant, and cotangent) for $\theta$ in this triangle?
3. How would these values change if the triangle were scaled by a factor?
4. Can you explain why rationalizing the denominator is often recommended?
5. What if $\theta$ were in a different quadrant; how would these ratios be affected?

Tip: Remember, in trigonometry, each ratio (sine, cosine, tangent) relates specific sides of the triangle, making them easy to remember if you know the definitions.