Problem 63 - Precalculus : Building Concepts and Connections

Question

Find the measure of the acute angle \(\theta\) for which the sine or cosine is given. $$\sin \theta=\frac{\sqrt{2}}{2}$$

Step-by-Step Solution

Answer

The acute angle \(\theta\) for which \(\sin \theta = \frac{\sqrt{2}}{2}\) is \(\frac{\pi}{4}\) radians.

Step 1: Recognize sine values for specific angles

The sine of an angle in the unit circle is equal to the y-coordinate of the point where the terminal side of the angle intersects the unit circle. Looking at the unit circle, it is noticed that when \(\theta = \frac{\pi}{4}\) radians, then \(\sin \theta = \frac{\sqrt{2}}{2}\). Also, the \(\theta\) value is acute.

Step 2: Write down the answer

As it was determined in the previous step, the acute angle \(\theta\) for which \(\sin \theta = \frac{\sqrt{2}}{2}\) is \(\frac{\pi}{4}\) radians.

Key Concepts

Trigonometric Functions

Trigonometric functions link angles to ratios, helping us explore relationships between angles and sides of triangles. There are six main trigonometric functions:

• Sine (\( \sin \theta \)
• Cosine (\( \cos \theta \)
• Tangent (\( \tan \theta \)
• Cosecant (\( \csc \theta \)
• Secant (\( \sec \theta \)
• Cotangent (\( \cot \theta \)

Understanding these functions is essential for solving problems involving angles and triangles, particularly in the context of the unit circle. The unit circle has a radius of one, simplifying the relationship between the angles and the functions. The sine function, of particular relevance to this exercise, gives the y-coordinate of a point on the unit circle corresponding to an angle \( \theta \), while the cosine function provides the x-coordinate. Knowing specific values, such as \( \sin \theta = \frac{\sqrt{2}}{2} \) at \( \theta = \frac{\pi}{4} \), can expedite solving trigonometric problems.

Acute Angles

Acute angles are angles that measure less than 90 degrees (or less than \( \frac{\pi}{2} \) radians when expressed in radians). These angles are small and appear in right triangles, where they are defined as the non-right angles.In the unit circle, every angle less than \( \frac{\pi}{2} \) radians is considered acute. This is significant because the trigonometric functions for these angles are positive in this part of the circle. Recognizing that \( \sin \theta = \frac{\sqrt{2}}{2} \) identifies an acute angle tells us it is located in the first quadrant, where both sine and cosine are positive. Knowing whether an angle is acute can help determine which quadrant you're dealing with and confirm the correctness of your trigonometric calculations.

Radians

Radians are a way of measuring angles that relate the arc length of a circle to its radius. Instead of degrees, which divide a circle into 360 parts, radians consider how many times a radius can wrap along the circumference of a circle.Understanding radians is crucial because they connect directly with the geometry of circles, making calculations with trigonometric functions more straightforward:

• A full circle equals \( 2\pi \) radians
• A right angle equals \( \frac{\pi}{2} \) radians
• An angle of \( \frac{\pi}{4} \) radians is \( 45^{\circ} \)

These relationships are especially handy when working with the unit circle. For example, knowing that \( \sin \theta = \frac{\sqrt{2}}{2} \) corresponds to \( \theta = \frac{\pi}{4} \) radians allows you to identify this common angle quickly. Mastering radians will enable you to solve calculus and physics problems with greater ease and accuracy.