Problem 44 - Algebra 2

Question

CHALLENGE If \(\tan \beta=\frac{3}{4},\) find \(\frac{\sin \beta \sec \beta}{\cot \beta}\)

Step-by-Step Solution

Answer

The value of \( \frac{\sin \beta \sec \beta}{\cot \beta} \) is \( \frac{9}{16} \).

Step 1: Understand the Equation

We need to find \( \frac{\sin \beta \sec \beta}{\cot \beta} \). It involves trigonometric identities: \( \sin \beta \), \( \sec \beta \), and \( \cot \beta \). We are given that \( \tan \beta = \frac{3}{4} \).

Step 2: Express Sin and Cos in Terms of Tan

Recall the identity \( \tan \beta = \frac{\sin \beta}{\cos \beta} \), and adjust variables so that \( \sin \beta = 3k \) and \( \cos \beta = 4k \), where \( k \) is a common factor.

Step 3: Solve for Common Factor Using Pythagorean Identity

Using \( \sin^2\beta + \cos^2\beta = 1 \), we have: \( (3k)^2 + (4k)^2 = 1 \). This gives \( 9k^2 + 16k^2 = 1 \), so \( 25k^2 = 1 \). Thus, \( k = \frac{1}{5} \).

Step 4: Determine \( \sin \beta \) and \( \cos \beta \)

Substitute \( k = \frac{1}{5} \) back: \( \sin \beta = 3k = \frac{3}{5} \) and \( \cos \beta = 4k = \frac{4}{5} \).

Step 5: Use Sin and Cos to Find Sec and Cot

We calculate \( \sec \beta = \frac{1}{\cos \beta} = \frac{5}{4} \) and \( \cot \beta = \frac{1}{\tan \beta} = \frac{4}{3} \).

Step 6: Calculate \( \frac{\sin \beta \sec \beta}{\cot \beta} \)

Plug in the values: \( \frac{\sin \beta \sec \beta}{\cot \beta} = \frac{\left( \frac{3}{5} \right) \times \left( \frac{5}{4} \right)}{\frac{4}{3}} \). This simplifies to \( \frac{\frac{15}{20}}{\frac{4}{3}} = \frac{3}{4} \times \frac{3}{4} = \frac{9}{16} \).

Key Concepts

Tangent

Tangent is one of the fundamental trigonometric functions. It relates to the angle of a right triangle with the lengths of the opposite side to the adjacent side. The tangent function can be defined as:

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

This equation expresses tangent in terms of sine and cosine.
Given \(tan \beta = \frac{3}{4}\), we can assume a right triangle where the opposite side is 3 and the adjacent side is 4.We find the hypotenuse using the Pythagorean theorem: \[ \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 \] This shows that the sides are in a 3-4-5 ratio.

Sine

The sine function is a key trigonometric function that compares the length of the side opposite the angle with the hypotenuse of a right triangle.Sine is expressed as:

• \( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \)

Using the triangle with \(\tan \beta = \frac{3}{4}\)and calculating the hypotenuse as 5, the sine of angle \( \beta \) is:\[ \sin \beta = \frac{3}{5} \]
These relationships provide strong connections between trigonometric functions, helping to solve various mathematical problems involving angles and measurements.

Cosine

Cosine is a trigonometric function that measures the ratio of the length of the adjacent side to the hypotenuse in a right triangle.The cosine function is described by:

• \( \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)

Following the given\(\tan \beta = \frac{3}{4}\), and previously identified hypotenuse, the cosine of angle \(\beta\) is:\[ \cos \beta = \frac{4}{5} \]
These ratios are not just numbers, they represent critical angles in mathematics, engineering, and physical sciences.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables.They provide elegant expressions to interrelate different functions, simplifying complex calculations.An important identity is the Pythagorean identity,\( \sin^2 \theta + \cos^2 \theta = 1 \).
For the given triangle:\[ (\sin \beta)^2 + (\cos \beta)^2 = (\frac{3}{5})^2 + (\frac{4}{5})^2 = 1 \]This confirms the relationship as true.Further simplifying calculations, the identities for secant and cotangent are:

• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)

These identities assist in reducing complex expressions and solving problems efficiently by offering alternative pathways to solutions.