Problem 19 - Basic Technical Mathematics with Calculus

Question

Find the values of the indicated functions. In Exercises \(17-20,\) give answers in exact form. In Exercises \(21-24,\) the values are approximate. Given \(\tan \theta=2,\) find \(\sin \theta\) and \(\sec \theta\).

Step-by-Step Solution

Answer

\( \sin \theta = \pm \frac{2}{\sqrt{5}} \) and \( \sec \theta = \pm \sqrt{5} \).

Step 1: Understand the Trigonometric Identity

We know that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Since we are given \( \tan \theta = 2 \), we can express this identity as \( \frac{\sin \theta}{\cos \theta} = 2 \). Our goal is to find \( \sin \theta \) and \( \sec \theta = \frac{1}{\cos \theta} \).

Step 2: Use Pythagorean Identity to Find Cos θ

We can use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \). From \( \tan \theta = 2 \), we have \( \sin \theta = 2 \cos \theta \). Substitute \( \sin \theta = 2 \cos \theta \) into the Pythagorean identity: \( (2 \cos \theta)^2 + \cos^2 \theta = 1 \).

Step 3: Simplify the Equation

Simplify the equation from the previous step: \( 4 \cos^2 \theta + \cos^2 \theta = 1 \), which simplifies further to \( 5 \cos^2 \theta = 1 \). Divide both sides by 5: \( \cos^2 \theta = \frac{1}{5} \).

Step 4: Solve for Cos θ and Sec θ

Take the square root of both sides to find \( \cos \theta \): \( \cos \theta = \pm \frac{1}{\sqrt{5}} \). Accordingly, \( \sec \theta = \frac{1}{\cos \theta} = \frac{\pm \sqrt{5}}{1} = \pm \sqrt{5} \).

Step 5: Find Sin θ Using Cos θ

Using \( \sin \theta = 2 \cos \theta \) and \( \cos \theta = \pm \frac{1}{\sqrt{5}} \), substitute to get \( \sin \theta = 2 \times \frac{1}{\sqrt{5}} = \frac{2}{\sqrt{5}} \).

Key Concepts

Trigonometric Identity

Trigonometric identities are equations that relate different trigonometric functions. These identities hold true for all values of the variables involved. They are fundamental tools in trigonometry, making complex calculations much simpler. One of the most commonly used identities is the one for tangent, expressed as:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

This identity tells us that the tangent of an angle is the ratio between the sine and cosine of that angle. Understanding and using this identity allows us to transform expressions and solve for unknown values of trigonometric functions when specific values are given. For example, if you're given \( \tan \theta = 2 \), you can cross-multiply to show that:

• \( \sin \theta = 2 \cos \theta \).

This equation becomes quite handy in conjunction with other identities to find the values of sine, cosine, and secant.

Pythagorean Identity

The Pythagorean identity is another essential relationship between trigonometric functions, given by:

• \( \sin^2 \theta + \cos^2 \theta = 1 \).

This identity is derived from the Pythagorean theorem, used in right-angled triangles, and is crucial for solving problems that involve trigonometric expressions. When you're given a specific value, such as \( \tan \theta = 2 \), you can use the equation \( \sin \theta = 2 \cos \theta \) from the trigonometric identity, substituting it into the Pythagorean identity as follows: Replace \( \sin \theta \) with \( 2 \cos \theta \):

• \( (2 \cos \theta)^2 + \cos^2 \theta = 1 \).

Simplifying, we get \( 4 \cos^2 \theta + \cos^2 \theta = 1 \) or \( 5 \cos^2 \theta = 1 \). Solving this equation helps us find the value of \( \cos \theta \), which remains instrumental in determining other trigonometric functions such as \( \sec \theta \).

Solve Equations

Solving trigonometric equations involves finding values that satisfy the equation for a variable, often ranging over angles. In our exercise, given \( \tan \theta = 2 \), we translate this into \( \sin \theta = 2 \cos \theta \) and employ trigonometric identities like the Pythagorean identity to progress in solving. Let’s break down the steps:

• Use the substitution \( \sin \theta = 2 \cos \theta \) in the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Substitute to form \( 4 \cos^2 \theta + \cos^2 \theta = 1 \).
• Solve for \( \cos^2 \theta \), yielding \( 5 \cos^2 \theta = 1 \) and further \( \cos^2 \theta = \frac{1}{5} \).
• Taking the square root gives \( \cos \theta = \pm \frac{1}{\sqrt{5}} \).
• Calculate \( \sec \theta = \frac{1}{\cos \theta} \), giving \( \pm \sqrt{5} \).
• Finally, find \( \sin \theta \) using \( \sin \theta = 2 \cos \theta \), resulting in \( \sin \theta = \frac{2}{\sqrt{5}} \).

This step-by-step method allows us to systematically determine the needed trigonometric values. By understanding and applying these principles, you can solve a wide range of trigonometric equations.