In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of
0, 𝜋/4, 𝜋/2, 3𝜋/4, 𝜋, 5𝜋/4, 3𝜋/2, 7𝜋/4, and 2𝜋.
a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function.
b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.
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sin 47𝜋/4

Verified step by step guidance

1. Step 1: Recognize that the problem involves evaluating \( \sin \left( \frac{47\pi}{4} \right) \) using the unit circle and periodic properties of the sine function.
2. Step 2: Recall that the sine function has a period of \( 2\pi \), meaning \( \sin(\theta) = \sin(\theta + 2k\pi) \) for any integer \( k \). Use this to reduce the angle \( \frac{47\pi}{4} \) to an equivalent angle between 0 and \( 2\pi \).
3. Step 3: To reduce the angle, subtract multiples of \( 2\pi \) (which is \( \frac{8\pi}{4} \)) from \( \frac{47\pi}{4} \) until the result lies within one full rotation (0 to \( 2\pi \)). This can be done by calculating \( \frac{47\pi}{4} - n \times 2\pi \) where \( n \) is an integer chosen to bring the angle into the principal range.
4. Step 4: Once the reduced angle \( \theta_{reduced} \) is found, identify its position on the unit circle divided into eight equal arcs (each arc corresponds to \( \frac{\pi}{4} \) radians). Use the known coordinates \( (x,y) \) for that angle to find \( \sin(\theta_{reduced}) = y \).
5. Step 5: Conclude that \( \sin \left( \frac{47\pi}{4} \right) = \sin(\theta_{reduced}) \) by the periodicity of sine, and use the coordinate from the unit circle to express the exact value.

Key Concepts

Unit Circle and Radian Measure

The unit circle is a circle with radius 1 centered at the origin, where angles are measured in radians. Each point on the circle corresponds to an angle t, with coordinates (cos t, sin t). Understanding how angles relate to points on the unit circle is essential for evaluating trigonometric functions like sine and cosine.

Periodic Properties of Trigonometric Functions

Trigonometric functions such as sine and cosine are periodic, meaning their values repeat at regular intervals. For sine and cosine, the period is 2π, so sin(t) = sin(t + 2πk) for any integer k. This property allows simplification of large angle measures by reducing them modulo 2π.

Reference Angles and Angle Reduction

Reference angles help find trigonometric values for angles outside the first rotation by relating them to acute angles within the first quadrant. By subtracting multiples of 2π or using symmetry, one can find equivalent angles with known sine or cosine values, facilitating the evaluation of functions at large or complex angles.