Problem 29 - Algebra 2

Question

Find the value of each expression. \(\tan \theta,\) if \(\cos \theta=\frac{4}{5} ; 0^{\circ} \leq \theta<90^{\circ}\)

Step-by-Step Solution

Answer

\( \tan \theta = \frac{3}{4} \).

Step 1: Use Pythagorean Identity

We know that for any angle \( \theta \) within a right triangle, the Pythagorean identity states that \( \sin^2\theta + \cos^2\theta = 1 \). Given \( \cos\theta = \frac{4}{5} \), we can substitute this into the identity: \( \sin^2\theta + \left( \frac{4}{5} \right)^2 = 1 \). Simplifying, we have \( \sin^2\theta + \frac{16}{25} = 1 \).

Step 2: Solve for \(\sin \theta\)

Subtract \( \frac{16}{25} \) from both sides of the equation to find \( \sin^2\theta \): \( \sin^2\theta = 1 - \frac{16}{25} \). Convert \( 1 \) to the fraction \( \frac{25}{25} \) to perform the subtraction: \( \sin^2\theta = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \). Solve for \( \sin\theta \) by taking the positive square root (since \( 0^{\circ} \leq \theta < 90^{\circ} \), \( \sin\theta \) is positive): \( \sin\theta = \frac{3}{5} \).

Step 3: Use the Definition of Tangent

The tangent of an angle \( \tan \theta \) is defined as the ratio of \( \sin \theta \) to \( \cos \theta \): \( \tan\theta = \frac{\sin\theta}{\cos\theta} \). Substitute the known values: \( \tan\theta = \frac{\frac{3}{5}}{\frac{4}{5}} \).

Step 4: Simplify the Expression for Tangent

To divide by a fraction, multiply by its reciprocal: \( \tan\theta = \frac{3}{5} \times \frac{5}{4} = \frac{3 \cdot 5}{5 \cdot 4} = \frac{3}{4} \). Therefore, \( \tan\theta = \frac{3}{4} \).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the foundational concepts in trigonometry. It states that for any angle \( \theta \), the sum of the squares of its sine and cosine is always equal to one:\[ \sin^2\theta + \cos^2\theta = 1 \]This is known as the Pythagorean Identity because it mirrors the Pythagorean Theorem, which relates the lengths of the sides of a right triangle. In our problem, we use this identity to solve for \( \sin \theta \) since we are given \( \cos \theta = \frac{4}{5} \). We substitute \( \cos \theta \) into the identity and simplify:

• Substitute: \( \sin^2\theta + \left( \frac{4}{5} \right)^2 = 1 \)
• Simplify: \( \sin^2\theta + \frac{16}{25} = 1 \)

By rearranging the equation to isolate \( \sin^2\theta \), we can find the value of \( \sin \theta \). This identity is essential in trigonometry because it allows us to find missing values and relate sine and cosine directly.

Tangent Function

The tangent function, \( \tan \theta \), is a primary trigonometric function that represents the ratio of the sine and cosine of an angle. In a right triangle, it is defined as the opposite side over the adjacent side. Mathematically, it is expressed as:\[ \tan\theta = \frac{\sin\theta}{\cos\theta} \]In our exercise, after finding \( \sin \theta = \frac{3}{5} \), we calculate \( \tan \theta \) by dividing \( \sin \theta \) by \( \cos \theta \):

• Given: \( \sin \theta = \frac{3}{5} \), \( \cos \theta = \frac{4}{5} \)
• \( \tan\theta = \frac{\frac{3}{5}}{\frac{4}{5}} \)

Divide the sine by cosine:

• This simplifies to: \( \tan\theta = \frac{3}{4} \)

Understanding the tangent function is vital as it shows how sine and cosine relate and allows us to find angles or sides in right triangles.

Sine and Cosine Relationship

In trigonometry, understanding the relationship between sine and cosine functions is crucial. These two functions are connected through the unit circle and trigonometric identities.- **Right Triangle Relation**: For a right triangle: - \( \sin \theta \) is the ratio of the length of the opposite side to the hypotenuse. - \( \cos \theta \) is the ratio of the length of the adjacent side to the hypotenuse.Using the Pythagorean Identity \( \sin^2\theta + \cos^2\theta = 1 \), we can derive one from the other, as shown in our problem:

• If \( \cos \theta = \frac{4}{5} \), you can find \( \sin \theta \).
• We calculated \( \sin^2\theta = \frac{9}{25} \), so \( \sin \theta = \frac{3}{5} \).

The unity relationship shows how changes in one function (like increasing \( \cos \theta \)) will affect \( \sin \theta \). This reciprocal adjustment keeps their combined square sum equal to one, underscoring their interdependence.