In the image, you have a right triangle with one of the acute angles labeled as $\theta$. The sides of the triangle are given as:

• Opposite side to $\theta$: 9
• Adjacent side to $\theta$: 12
• Hypotenuse: This can be calculated using the Pythagorean theorem.

Let me calculate the hypotenuse first:

$\text{Hypotenuse}^2 = 9^2 + 12^2 = 81 + 144 = 225$ $\text{Hypotenuse} = \sqrt{225} = 15$

Now that we know the hypotenuse is 15, we can evaluate the six trigonometric functions of $\theta$:

1. $\sin \theta$ (sine) is the ratio of the opposite side to the hypotenuse: $\sin \theta = \frac{9}{15} = \frac{3}{5}$

2. $\cos \theta$ (cosine) is the ratio of the adjacent side to the hypotenuse: $\cos \theta = \frac{12}{15} = \frac{4}{5}$

3. $\tan \theta$ (tangent) is the ratio of the opposite side to the adjacent side: $\tan \theta = \frac{9}{12} = \frac{3}{4}$

4. $\csc \theta$ (cosecant) is the reciprocal of sine: $\csc \theta = \frac{1}{\sin \theta} = \frac{5}{3}$

5. $\sec \theta$ (secant) is the reciprocal of cosine: $\sec \theta = \frac{1}{\cos \theta} = \frac{5}{4}$

6. $\cot \theta$ (cotangent) is the reciprocal of tangent: $\cot \theta = \frac{1}{\tan \theta} = \frac{4}{3}$

Summary:

• $\sin \theta = \frac{3}{5}$
• $\cos \theta = \frac{4}{5}$
• $\tan \theta = \frac{3}{4}$
• $\csc \theta = \frac{5}{3}$
• $\sec \theta = \frac{5}{4}$
• $\cot \theta = \frac{4}{3}$

Let me know if you want more details or have any questions!

Here are 5 related questions:

1. How do you derive the values of trigonometric functions for any right triangle?
2. How does the Pythagorean theorem apply in right triangle trigonometry?
3. What are the reciprocal identities in trigonometry?
4. How are trigonometric ratios used to solve real-life problems?
5. How can you find the angles of a triangle given only the sides?

Tip: Always remember the SOH-CAH-TOA mnemonic to quickly recall the basic trigonometric ratios.