Problem 5 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Find \(\tan \theta\) if \(\sin \theta=\frac{4}{5}\) and \(0 \leq \theta \leq \frac{\pi}{2}\).

Step-by-Step Solution

Answer

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

Step 1: Understand the given information

The exercise provides \(\sin \theta = \frac{4}{5}\) and \(0 \leq \theta \leq \frac{\pi}{2}\), meaning \(\theta\) is in the first quadrant where all trigonometric functions are positive.

Step 2: Use the Pythagorean identity

Recall the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute \(\sin \theta = \frac{4}{5}\) into the identity: \( (\frac{4}{5})^2 + \cos^2 \theta = 1 \).

Step 3: Solve for \(\cos \theta\)

Calculate \((\frac{4}{5})^2 = \frac{16}{25}\). Then solve for \(\cos^2 \theta\) by rearranging the equation: \(\cos^2 \theta = 1 - \frac{16}{25} = \frac{9}{25}\). Finally, find the square root: \(\cos \theta = \frac{3}{5}\).

Step 4: Find \(\tan \theta \)

Use the definition \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substitute the values: \(\tan \theta = \frac{\frac{4}{5}}{\frac{3}{5}}\).

Step 5: Simplify

Simplify the expression: \(\tan \theta = \frac{4}{5} \times \frac{5}{3} = \frac{4}{3}\).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry. It states that for any angle \theta, the square of the sine function plus the square of the cosine function is always equal to 1. Mathematically, this is written as: \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity is extremely useful for finding unknown trigonometric functions when one function is known. For example, if we know \( \sin \theta = \frac{4}{5} \), we can find \( \cos \theta \) by rearranging the identity: \( \cos^2 \theta = 1 - \sin^2 \theta \.\). Substituting the value, \( \cos^2 \theta = 1 - \( \frac{4}{5} \)^2 = 1 - \frac{16}{25} = \frac{9}{25} \.\) By taking the square root, we get \( \cos \theta = \frac{3}{5} \.\) Understanding the Pythagorean identity is essential for solving many trigonometric problems.

Sine Function

The sine function, written as \( \sin \theta \,\), is a trigonometric function that represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. In this exercise, we are given \( \sin \theta = \frac{4}{5} \.\) This means that if we form a right triangle, the side opposite the angle \( \theta \) has a length of 4 units, and the hypotenuse has a length of 5 units. The sine function is always between -1 and 1. Knowing \(\sin \theta\) allows us to easily calculate other trigonometric functions like cosine and tangent.

Tangent Function

The tangent function, represented as \( \tan \theta \,\), is defined as the ratio of the sine function to the cosine function: \( \tan \theta = \frac{ \sin \theta}{\cos \theta} \.\) In this exercise, once we have \( \sin \theta = \frac{4}{5} \,\) and \( \cos \theta = \frac{3}{5} \,\) we can find the tangent function by substituting the values: \( \tan \theta = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \.\) The tangent function provides the ratio of the opposite side to the adjacent side in a right-angled triangle. It's especially useful when working with angles in right triangles.

First Quadrant

The first quadrant refers to the section of the coordinate plane where both x and y values are positive. In trigonometry, angles in the first quadrant range from \0 \leq \theta \leq \frac{\pi}{2} \.\) All trigonometric functions - sine, cosine, and tangent - are positive in this quadrant. This is important because the exercise specifies that \( \theta \) is in the first quadrant. Therefore, we don't need to worry about negative values for \( \sin \,\ \( \cos \), and \( \tan \.\) Understanding the quadrant helps in knowing the sign of the functions and provides a clearer picture of where the angle lies on the unit circle.