Find the value of the six trigonometric functions of the angle theta in the figure.
4
StartRoot 3 EndRoot
theta
A
B
C
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Part 1
For the given​ triangle, which of the following ratios represents the trigonometric function sine theta​?
A.StartFraction AB Over AC EndFraction
StartFraction AB Over AC EndFraction
B.StartFraction AC Over AB EndFraction
StartFraction AC Over AB EndFraction
C.StartFraction BC Over AC EndFraction
StartFraction BC Over AC EndFraction
D.StartFraction AB Over BC EndFraction
StartFraction AB Over BC EndFraction
E.StartFraction AC Over BC EndFraction
StartFraction AC Over BC EndFraction
F.StartFraction BC Over AB EndFraction
StartFraction BC Over AB EndFraction

To determine which ratio represents the trigonometric function sine \(\theta\), let's consider the sides of the right triangle given in the problem:

- Let's assume that \(AB\) is the side opposite to the angle \(\theta\).
- \(BC\) is the side adjacent to the angle \(\theta\).
- \(AC\) is the hypotenuse of the triangle.

The sine of an angle in a right triangle is given by the ratio of the length of the side opposite the angle to the length of the hypotenuse:

\[
\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}
\]

Given this, the correct ratio that represents \(\sin \theta\) is:

\[
\sin \theta = \frac{AB}{AC}
\]

So, the correct answer is:

**A. \(\frac{AB}{AC}\)**

Would you like further details or explanations? Here are some related questions:

1. What are the definitions of the other five trigonometric functions?
2. How do you determine the hypotenuse in a right triangle?
3. How do you calculate the value of \(\sin \theta\) if the lengths of \(AB\) and \(AC\) are given?
4. What is the relationship between sine and cosine in a right triangle?
5. Can you derive the other trigonometric functions if \(\sin \theta\) is known?

**Tip:** In a right triangle, always remember the mnemonic SOHCAHTOA to recall the basic trigonometric ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent.

Find the value of the six trigonometric functions of the angle theta in the figure. For the given triangle, which of the following ratios represents the trigonometric function sine theta?

Explore how to determine the sine of angle theta using the given right triangle. This problem guides you through identifying trigonometric ratios, applying the SOHCAHTOA mnemonic, and calculating the sine, cosine, and tangent functions. Perfect for students in Grades 9-10.

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