Problem 28 - 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. $$ 135^{\circ} $$

Step-by-Step Solution

Answer

\(\sin(135^{\circ}) = \sqrt{2}/2 \approx 0.71\), \(\cos(135^{\circ}) = -\sqrt{2}/2 \approx -0.71\).

Step 1: Determine Exact Value of Sine and Cosine

Since 135 degrees is in the second quadrant where sine is positive and cosine is negative, we anticipate respective positive and negative values. By recalling the special right triangles or the unit circle, we know that \(\sin(135^{\circ}) = \sqrt{2}/2\) and \(\cos(135^{\circ}) = -\sqrt{2}/2\).

Step 2: Convert Exact Values to Decimal

Next, we use a calculator to obtain the decimal equivalents. So, \(\sin(135^{\circ}) = \sqrt{2}/2 \approx 0.71\) and \(\cos(135^{\circ}) = -\sqrt{2}/2 \approx -0.71\). Our final answers are therefore \(\sin(135^{\circ}) \approx 0.71\) and \(\cos(135^{\circ}) \approx -0.71\).

Key Concepts

Sine and Cosine

Sine and cosine are two of the most fundamental functions in trigonometry, and they are vital for understanding angles and measurement in mathematics. These functions stem from the relationship between an angle and a right triangle.

• The sine of an angle is the ratio of the length of the side opposite the angle to the hypotenuse.
• The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.

The sine and cosine values help describe the shape and size of right triangles and are also used in wave functions and oscillations. Understanding these concepts through triangles or the unit circle enables us to find the sine and cosine of various angles, including non-standard ones like 135°. Knowing where these angles are located on the unit circle helps anticipate whether they might be positive or negative, depending on the quadrant.

Unit Circle

The unit circle is an invaluable tool in trigonometry. It's a circle with a radius of 1, centered at the origin of the coordinate plane. By using the unit circle, we can easily determine sine and cosine values for any angle. The key idea is to link angles to coordinates on the circle.

• The x-coordinate of a point on the unit circle corresponds to the cosine of an angle.
• The y-coordinate corresponds to the sine of the angle.

For example, the angle 135° is measured counterclockwise from the positive x-axis. It corresponds to the coordinates \((-rac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\)\) on the unit circle. That's how we know that \(\cos(135^{\circ}) = -\frac{\sqrt{2}}{2}\) and \(\sin(135^{\circ}) = \frac{\sqrt{2}}{2})." This insight is especially useful for understanding angles that aren't found by just looking at simple right triangles.

Quadrants

Quadrants divide the coordinate plane into four sections, each of which has a unique combination of signs for sine and cosine values. Knowing the quadrant of an angle quickly tells you something vital about the sine and cosine of that angle:

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, cosine is negative.
• Quadrant III: Both are negative.
• Quadrant IV: Sine is negative, cosine is positive.

The angle 135° is situated in the second quadrant, where, as the rule goes, the sine value is positive and the cosine value is negative. This information is particularly handy for determining the sign of trigonometric functions immediately, even before calculating the exact values.

Special Right Triangles

Special right triangles, specifically the 45°-45°-90° triangle, are a key geometric concept in trigonometry. These triangles provide foundational building blocks for understanding trigonometric ratios.

• In a 45°-45°-90° triangle, both non-hypotenuse sides are of equal length.
• The hypotenuse is \(\sqrt{2}\) times as long as each leg.

Knowing these ratios allows quick calculation of trigonometric values for multiples of 45°. For example, an angle of 135° can be related to 45° by understanding it as \(180° - 45°\), placing it in the second quadrant. By remembering the proportions of this special right triangle, we can instantly know that \(\sin(135°) = \frac{\sqrt{2}}{2}\) and \(\cos(135°) = -\frac{\sqrt{2}}{2}\). Mastering these special triangles simplifies the process of figuring out angles' sine and cosine values substantially.