Question

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

Short answer

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

Step-by-step solution

Understand Coterminal Angles

Coterminal angles are angles that differ by a full rotation (360°). To find a coterminal angle, you can either add or subtract 360°.

Find a Coterminal Angle Between 0° and 360°

To simplify the angle to a value between 0° and 360°, subtract 360° from 420°: \[ 420^{\text{°}} - 360^{\text{°}} = 60^{\text{°}} \]. So, 420° is coterminal with 60°.

Use the Coterminal Angle

Since 420° is coterminal with 60°, we can use the trigonometric properties of 60° to find the secant. The secant function is the reciprocal of the cosine function, so \[ \text{sec} \theta = \frac{1}{\text{cos} \theta} \].

Find the Cosine of the Coterminal Angle

The cosine of 60° is \[ \text{cos} 60^{\text{°}} = \frac{1}{2} \].

Calculate the Secant

Using the definition of secant, \[ \text{sec} 60^{\text{°}} = \frac{1}{\text{cos} 60^{\text{°}}} = \frac{1}{\frac{1}{2}} = 2 \].

Key concepts

Secant Function

The secant function, denoted as \text{sec}\(\theta\), is one of the basic trigonometric functions. It represents the reciprocal of the cosine function. In mathematical terms, this is written as \text{sec}\(\theta\) = \frac{1}{\text{cos}\(\theta\)}. This function is used to relate the angle \theta\ to a ratio derived from a right triangle's hypotenuse and adjacent side. The secant function is particularly useful when dealing with angles and their properties in various applications such as engineering, physics, and geometry.

Trigonometric Properties

Understanding trigonometric properties is vital to solving many problems involving angles. Trigonometric functions, such as sine, cosine, and secant, have specific values and identities that help in simplifying and calculating these problems. For example, the value of \text{cos}\(60\text{°}\) is always \frac{1}{2}\, regardless of the context. Knowing these properties allows for quick and accurate computation of different functions. Additionally, these properties help in relating different angles and their corresponding ratios, making complex trigonometric relationships more manageable.

Cosine Function

The cosine function, denoted as \text{cos}\(\theta\), measures the adjacent side over the hypotenuse in a right-angled triangle. This function is essential for understanding many aspects of trigonometry. For instance, in the exercise, the cosine of 60° is used to find the secant of 420°. The value \text{cos}\(60\text{°}\) = \frac{1}{2}\ is derived from the unit circle and is a standard trigonometric value. By understanding the cosine function, we can derive the secant function and simplify complex angles into their more understandable coterminal forms.

Angle Simplification

Angle simplification deals with converting complex angles into more manageable, equivalent angles known as coterminal angles. An angle is coterminal with another if it differs by a full rotation of 360°. For instance, 420° can be simplified by subtracting 360°, giving us 60°, which is easier to work with. This concept is crucial in trigonometry because it allows for the simplification of expressions, making calculations more straightforward. To find coterminal angles, you simply add or subtract multiples of 360° until the angle falls within the 0° to 360° range.