Problem 13 - Algebra 2

Question

Find the exact value of each function. \(\sin 690^{\circ}\)

Step-by-Step Solution

Answer

The exact value of \(\sin 690^{\circ}\) is \(-\frac{1}{2}\).

Step 1: Understanding the Angle

The angle given is \[ 690^{\circ} \].This angle is greater than \(360^{\circ}\), so it makes sense to find an equivalent angle within the first cycle of the unit circle, \([0^{\circ}, 360^{\circ})\). This can be done by subtracting multiples of \(360^{\circ}\).

Step 2: Reduce the Angle

Determine how many full rotations of \(360^{\circ}\) can be subtracted from \[ 690^{\circ} \].Since \[ 690^{\circ} - 360^{\circ} = 330^{\circ}, \]an equivalent angle is \[ 330^{\circ} \].It indicates that\( \sin 690^{\circ} = \sin 330^{\circ} \).

Step 3: Evaluate the Sine Function at 330°

Now, look for the sine of \( 330^{\circ} \).Since \( 330^{\circ} \) is in the fourth quadrant, where the sine is negative and considering the reference angle \( 360^{\circ} - 330^{\circ} = 30^{\circ}, \)the value is \( -\frac{1}{2}. \)

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1 centered at the origin of a coordinate system. The unit circle allows us to easily evaluate trigonometric functions like sine, cosine, and tangent for any angle.

All angles on the unit circle are measured from the positive x-axis, moving counter-clockwise. This makes understanding and identifying angle positions straightforward. Each point on the unit circle corresponds to

• an angle measured in degrees or radians,
• a coordinate pair consisting of \(x\) and \(y\) values,
• these coordinates directly relate to the cosine and sine of the angle.

For example, an angle \( \theta\) on the unit circle will have coordinates \((\cos \theta, \sin \theta)\).

By knowing these coordinates, you can easily determine the value of trigonometric functions and understand the symmetry and periodicity inherent in trigonometry.

Reference Angles

Reference angles are key to simplifying the computation of trigonometric functions. A reference angle is the smallest acute angle a given angle makes with the x-axis. Understanding reference angles can help you find the values of trigonometric functions for non-standard angles.

To find the reference angle:

• For angles in the first quadrant, the reference angle is the angle itself.
• In the second quadrant, it's \(180^\circ - \theta\).
• For the third quadrant, you calculate \((\theta - 180^\circ)\).
• In the fourth quadrant, it's \(360^\circ - \theta\).

In the case of finding \(\sin 690^\circ\), the angle is equivalent to \(330^\circ\) when reduced, and \(330^\circ\) has a reference angle of \(30^\circ\).

Reference angles allow the determination of trigonometric function values based on simpler, commonly memorized angles like \(0^\circ, 30^\circ, 45^\circ, 60^\circ, \) and \(90^\circ\).

Quadrants

The coordinate system is divided into four quadrants, each one formed by the intersection of the x-axis and y-axis. Understanding which quadrant an angle falls into helps determine the sign of its trigonometric functions.

The quadrants are labeled as follows:

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive, while cosine and tangent are negative.
• Quadrant III: Tangent is positive, while sine and cosine are negative.
• Quadrant IV: Cosine is positive, but sine and tangent are negative.

In our example of \(\sin 690^\circ\), we reduced this angle to \(330^\circ\) which is in the fourth quadrant. Here, sine values are negative because only cosine remains positive.

This quadrant understanding is crucial when determining the exact value of trigonometric functions for any angle, helping predict whether those values will be positive or negative once calculated.