Question

True or False. If \(\theta\) is an acute angle and \(\sec \theta=3\), then \(\cos \theta=\frac{1}{3}\)

Short answer

True

Step-by-step solution

Identify the given information

You are given that \(\theta\) is an acute angle and \(\sec \theta = 3\).

Understand the relationship between secant and cosine

Recall that \(\sec \theta\) is the reciprocal of \(\cos \theta\), which means \[\sec \theta = \frac{1}{\cos \theta}\]

Set up the equation

Using the relationship, you can write \[\frac{1}{\cos \theta} = 3\]

Solve for \(\cos \theta\)

To find \(\cos \theta\), take the reciprocal of both sides: \[\frac{1}{3} = \cos \theta\]

Evaluate the statement

Now compare the result with the given statement. Since \(\cos \theta = \frac{1}{3}\), the initial statement is true.

Key concepts

acute angle

An angle is considered acute if its measurement falls between 0 and 90 degrees. These angles are significant in trigonometry as they often appear in the context of right triangles and unit circles. In the given problem, the angle θ is specified to be acute. This is important because it ensures that the cosine and secant values are positive, which affects how we interpret \(\theta\) and \(\text{sec}\ \theta \). When \(\theta\) is acute, trigonometric functions such as cosine and secant will yield positive results, simplifying calculations and interpretations.

secant and cosine relationship

Secant \(\text{sec}\ \theta\) and cosine \( \text{cos}\ \theta\) are closely related in trigonometry. Specifically, secant is the reciprocal of cosine. This relationship can be expressed mathematically as follows:
\[\sec\ \theta = \frac{1}{\cos\ \theta}\]
This means that if you know the value of \(\sec\ \theta\), you can find \( \cos\ \theta\) by taking the reciprocal of secant. In the problem, it’s given that \(\text{sec}\ \theta = 3\). Using the reciprocal identity, you can find:
\[\cos\ \theta = \frac{1}{\sec\ \theta} = \frac{1}{3}\]
This straightforward relationship makes solving for one function easier when you know the other.

reciprocal identities

Reciprocal identities in trigonometry are fundamental relationships that express trigonometric functions in terms of their reciprocals. For the primary trigonometric functions—sine, cosine, and tangent—there are corresponding reciprocal identities:

• Sine (sin) and cosecant (csc): \[\csc\ \theta = \frac{1}{\sin\ \theta} \]
• Cosine (cos) and secant (sec): \[\sec\ \theta = \frac{1}{\cos\ \theta} \]
• Tangent (tan) and cotangent (cot): \[\cot\ \theta = \frac{1}{\tan\ \theta} \]

These identities are useful for solving various trigonometric problems because they provide alternative ways to express and manipulate equations. In the given problem, we directly used the reciprocal relationship between secant and cosine. Understanding reciprocal identities can broaden your ability to solve and simplify trigonometric expressions.