Problem 33 - Basic Technical Mathematics with Calculus

Question

Answer the given questions. If \(\tan \theta=3 / 4,\) what is the value of \(\sin ^{2} \theta+\cos ^{2} \theta ?\) \(\left[\sin ^{2} \theta=(\sin \theta)^{2}\right]\)

Step-by-Step Solution

Answer

The value is 1.

Step 1: Understanding the Problem

We are given that \( \tan \theta = \frac{3}{4} \) and need to find the value of \( \sin^2 \theta + \cos^2 \theta \).

Step 2: Recognizing a Trigonometric Identity

Recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity holds true for any angle \( \theta \).

Step 3: Applying the Identity

Recognizing that \( \sin^2 \theta + \cos^2 \theta = 1 \) based on the Pythagorean identity, there is no further computation needed, irrespective of knowing \( \tan \theta = \frac{3}{4} \).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental principle in trigonometry. It reveals the intrinsic relationship between the sine and cosine functions. This identity states that for any angle \( \theta \): \[ \sin^2 \theta + \cos^2 \theta = 1 \]This identity directly derives from the Pythagorean theorem, which applies to right triangles. Here, the square of the hypotenuse is equal to the sum of the squares of the other two sides. By interpreting the unit circle, where the radius (or hypotenuse) is 1, we can see this relationship come to life:- The horizontal leg is \( \cos \theta \).- The vertical leg is \( \sin \theta \).When these are squared and summed, they equal the radius squared, which is always 1. This powerful identity helps in simplifying many trigonometric problems and is the foundation for other trigonometric identities and formulas.

Sine and Cosine

The sine and cosine functions are two of the most essential in trigonometry. They are defined using the unit circle, where a circle has a radius of 1 centered at the origin of a coordinate plane.- **Sine Function**: For an angle \( \theta \), the sine value is the y-coordinate of the point on the unit circle. This can be expressed as \( \sin \theta \).- **Cosine Function**: Similarly, the cosine value is the x-coordinate of the point on the unit circle, expressed as \( \cos \theta \).These functions are periodic, meaning they repeat their values in regular intervals. - Their period is \(2\pi\) for one full revolution around the circle.- Both sine and cosine values range between -1 and 1.Understanding these basic definitions allows one to grasp their applications in calculating angles and modeling wave patterns, among other uses in mathematics and physics.

Tangent Function

The tangent function might initially seem a bit tricky, but it can be understood easily by its relationship with the sine and cosine functions.- It is defined as the ratio of the sine function to the cosine function: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]This means that tangent essentially "compares" the sine value against the cosine value for a particular angle \( \theta \). This is periodical as well, repeating its values every \(\pi\).- One important thing to note is that the tangent function is undefined whenever \( \cos \theta = 0 \) because division by zero is undefined. At these points, the tangent function has vertical asymptotes.Knowing \( \tan \theta = \frac{3}{4} \), as given in the exercise, enables you to find interesting properties or ratios in specific triangles, especially useful in geometry and trigonometry to determine side lengths and angles.