Question

Use a coterminal angle to find the exact value of each expression. Do not use a calculator. $$ \cos 420^{\circ} $$

Short answer

The exact value of \( \text{cos}(420^{\text{°}}) \) is \( \frac{1}{2} \).

Step-by-step solution

Understand Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but may have different measures. To find coterminal angles, you can add or subtract multiples of 360° from the given angle.

Find a Coterminal Angle

To find a coterminal angle within the standard range (0° to 360°), subtract 360° from 420°: \[ 420^{\text{°}} - 360^{\text{°}} = 60^{\text{°}} \] So, 60° is a coterminal angle of 420°.

Find the Cosine of the Coterminal Angle

Now, find the cosine of the coterminal angle 60°. \[ \text{cos}(60^{\text{°}}) = \frac{1}{2} \]

Key concepts

Finding Coterminal Angles

Coterminal angles are angles that share the same terminal side. That means they look identical when drawn in the standard position on the coordinate plane. To find coterminal angles, you can add or subtract multiples of 360°, because a full circle measures 360°.

For example, if you have an angle of 420°, you can subtract 360° to find a coterminal angle: 420° - 360° = 60°

So, 420° is coterminal with 60°. Both angles point in the same direction, even though their measurements are different. This helps when you need to find trigonometric values, like cosine or sine, because the trigonometric functions will have the same values for coterminal angles.

Cosine Function

The cosine function is one of the basic trigonometric functions and is denoted as \(\text{cos}\). It represents the x-coordinate of a point on the unit circle. Cosine is very useful for measuring the horizontal distance of a point on the circle from the origin.

To visualize this, imagine a unit circle centered at the origin of a coordinate system. If you draw a line from the origin to any point on this circle, the cosine of the angle \( \theta\) formed by this line with the positive x-axis is equal to the x-coordinate of the point where the line intersects the circle.

For example, for an angle of 60° (or 420°, which is coterminal with 60°), the cosine is found at the x-coordinate of the point on the circle where this line intersects. This point is at \(\frac{1}{2}, \frac{\text{3}}{\text{2}}\), so \( \text{cos}(60^{\text{°}}) = \frac{1}{2}\).

Exact Trigonometric Values

Exact trigonometric values are the precise values of trigonometric functions for certain special angles. These values do not require a calculator and are critical for solving many trigonometry problems.

For 60° (and its coterminal angles like 420°), the exact trigonometric values are well-known:

• \( \text{sin}(60°) = \frac{\text{sqrt(3)}}{2} \)
• \( \text{cos}(60°) = \frac{1}{2} \)
• \( \text{tan}(60°) = \text{sqrt(3)} \)

To find these values without a calculator, you can use triangle properties or the unit circle. Remember, knowing these exact values will help you solve problems quickly and accurately. For the problem given, we easily found that \( \text{cos}(420^{\text{°}}) = \text{cos}(60^{\text{°}}) = \frac{1}{2} \), which is an exact value.