Evaluate each expression without using a calculator.
sin (arccos (3/4))

Verified step by step guidance

1. Recognize that \( \arccos(3/4) \) represents an angle \( \theta \) such that \( \cos(\theta) = \frac{3}{4} \).
2. Visualize or draw a right triangle where the adjacent side to angle \( \theta \) is 3 and the hypotenuse is 4.
3. Use the Pythagorean theorem to find the opposite side: \( a^2 + 3^2 = 4^2 \), where \( a \) is the length of the opposite side.
4. Solve for \( a \) to find the length of the opposite side: \( a = \sqrt{4^2 - 3^2} \).
5. Now, use the definition of sine: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{4} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccos, are used to find the angle whose cosine is a given value. For example, arccos(3/4) gives the angle θ such that cos(θ) = 3/4. Understanding how to interpret these functions is crucial for evaluating expressions involving them.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This identity is essential when working with trigonometric functions, as it allows us to find the sine of an angle if we know its cosine. In this case, knowing cos(θ) = 3/4 helps us calculate sin(θ) using this identity.

Trigonometric Ratios

Trigonometric ratios relate the angles of a right triangle to the lengths of its sides. For a given angle θ, sin(θ) is defined as the ratio of the length of the opposite side to the hypotenuse. By applying these ratios, we can derive the sine value from the cosine value obtained from the inverse function.