Problem 26 - Algebra 2

Question

Find the exact values of the cosine and sine of each angle. Then find the decimal values. Round your answers to the nearest hundredth. $$ 315^{\circ} $$

Step-by-Step Solution

Answer

The exact values of cosine and sine for \(315^\circ\) are \(\frac{\sqrt{2}}{2}\) and \(-\frac{\sqrt{2}}{2}\) respectively. In decimal form, both values are approximated to 0.71 and -0.71 respectively.

Step 1: Express the angle in terms of π

The angle can be expressed in terms of π by multiplying it by \(\frac{\pi}{180}\) to convert it from degrees to radians. So, \(315^\circ = \frac{315\pi}{180} = \frac{7\pi}{4}\) radians.

Step 2: Identify the quadrant where the angle lies

By observing the angle, it can be seen that the angle \(\frac{7\pi}{4}\) lies in the fourth quadrant. In this quadrant, cosine is positive and sine is negative.

Step 3: Find the reference angle

The reference angle is the acute angle that the terminal side of the given angle forms with the x-axis. Since \(\frac{7\pi}{4}\) lies in the fourth quadrant, the reference angle is \(\frac{7\pi}{4} - 2\pi = \frac{\pi}{4}\). Note that \(2\pi\) is subtracted because it represents a full circle.

Step 4: Calculate the values of cosine and sine

The cosine and sine of a reference angle are equal to the cosine and sine of the given angle. Therefore, the cosine of \(315^\circ\) is equal to the cosine of \(\frac{\pi}{4}\) which is \(\frac{\sqrt{2}}{2}\), and the sine of \(315^\circ\) is equal to negative sine \(\frac{\pi}{4}\) which is \(-\frac{\sqrt{2}}{2}\) in the fourth quadrant.

Step 5: Convert the values into decimal form

The decimal form of \(\frac{\sqrt{2}}{2}\) and \(-\frac{\sqrt{2}}{2}\) can be found by performing the division and then rounding to the nearest hundredth. So, cosine \(315^\circ\) = 0.71 and sine \(315^\circ\) = -0.71.

Key Concepts

Cosine and Sine

Cosine and sine are fundamental trigonometric functions representing the ratios of sides in a right-angled triangle. They are used to determine the horizontal and vertical positions on a unit circle, respectively. These functions help us relate angles to ratios:

• Cosine (\( \cos \theta \)): Tracks the x-coordinate, or horizontal value, on the unit circle.
• Sine (\( \sin \theta \)): Tracks the y-coordinate, or vertical distance, on the unit circle.

By understanding these functions, we can find the exact and decimal values of trigonometric expressions. For instance, for the angle \( 315^{\circ} \), both sine and cosine can be extracted from known reference angles on the unit circle and then converted to decimals for ease of use and comprehension.

Angle Conversion

Converting angles between degrees and radians is essential in trigonometry. Degrees are often used in practical settings, while radians are the standard in mathematical calculations. The conversion is seamless using the formula:

• For degrees to radians: Multiply by \( \frac{\pi}{180} \)
• For radians to degrees: Multiply by \( \frac{180}{\pi} \)

For example, to convert \( 315^{\circ} \) to radians, we do the following: \[ 315^{\circ} \times \frac{\pi}{180} = \frac{7\pi}{4} \] radians. The conversion is crucial for identifying angle positions and making further calculations in various trigonometric contexts.

Reference Angle

A reference angle is the acute angle formed by the terminal side of an angle with the x-axis. It is never more than \( 90^{\circ} \) or \( \frac{\pi}{2} \). The reference angle helps in simplifying the calculation of trigonometric functions as it provides an equivalent angle in the first quadrant, where values are most familiar.When determining the reference angle for \( 315^{\circ} \), first identify the quadrant. Since it is in the fourth quadrant, subtract \( 2\pi \) from \( \frac{7\pi}{4} \) to get:\[ \frac{7\pi}{4} - 2\pi = \frac{-\pi}{4} \]Here, we take the absolute value in calculating trigonometric ratios, resulting in the reference angle being \( \frac{\pi}{4} \). This simplifies finding exact cosine and sine values, as the same reference angles are used across different quadrants with corresponding sign changes.

Quadrants in Trigonometry

The unit circle is divided into four distinct quadrants, each influencing the sign and values of cosine and sine based on the angle's position:

• First Quadrant (0 to\( 90^{\circ} \)): Both sine and cosine are positive.
• Second Quadrant (90^{\circ} to\( 180^{\circ} \)): Sine is positive, cosine is negative.
• Third Quadrant (180^{\circ} to\( 270^{\circ} \)): Both sine and cosine are negative.
• Fourth Quadrant (270^{\circ} to\( 360^{\circ} \)): Cosine is positive, sine is negative.

For \( 315^{\circ} \), it falls in the fourth quadrant. Here, cosine values remain positive (\( \cos(\frac{7\pi}{4}) = \frac{\sqrt{2}}{2} \)), and sine values are negative (\( \sin(\frac{7\pi}{4}) = -\frac{\sqrt{2}}{2} \)). Understanding these quadrant rules helps predict the sign of trigonometric functions for any given angle, aiding in accurate calculations and predictions.