Problem 82 - Algebra 2

Question

For Exercises \(79-82,\) suppose tan \(\theta=\frac{4}{3}, \sin \theta>0,\) and \(-\frac{\pi}{2} \leq \theta<\frac{\pi}{2}\) . Enter each answer as a decimal. What is \(\sin \theta+\cos \theta+\cot \theta+\csc \theta ?\)

Step-by-Step Solution

Answer

The requested expression \(\sin \theta + \cos \theta + \cot \theta + \csc \theta = 2.2\).

Step 1: Calculate sin θ and cos θ

Given tan θ = \frac{4}{3}, we know that tan θ = \frac{Sin \theta}{Cos \theta} = \frac{4}{3}. As Sin θ is given to be positive and tan θ is positive, we can deduce that Cos θ is also positive. Therefore, sin θ = \frac{4}{5} and cos θ = \frac{3}{5} are obtained by forming a right triangle with opposite side as 4, adjacent side as 3 and hypotenuse as 5 (by Pythagoras theorem).

Step 2: Calculate cot θ and csc θ

As cot θ is the reciprocal of tan θ, so cot θ = \frac{1}{tan \theta} = \frac{3}{4}. Similarly, csc θ as the reciprocal of sin θ, so csc θ = \frac{1}{sin \theta} = \frac{5}{4}.

Step 3: Subtract and calculate the expression

Substitute these values back into the main equation: \(\sin \theta + \cos \theta + \cot \theta + \csc \theta = \frac{4}{5} + \frac{3}{5} + \frac{3}{4} + \frac{5}{4} = 2.2\).

Key Concepts

Tangent Function

The tangent function is one of the primary trigonometric functions. It represents the ratio of the opposite side to the adjacent side in a right triangle. Given by \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), it links the sine and cosine functions.
When \( \tan \theta = \frac{4}{3} \), it implies that the opposite side of the triangle is 4 units, and the adjacent side is 3 units. This ratio helps us understand angles and their relationships in the context of geometry.
The value of \( \tan \theta \) being positive indicates that both sine and cosine are either both positive or both negative, maintaining their respective sign in that specific quadrant, which in this scenario is the first quadrant as both \( \sin \theta \) and \( \cos \theta \) are positive. This helps in solving trigonometric expressions by providing specific constraints and solutions.

Sine and Cosine

Sine and cosine are foundational trigonometric functions that describe the relationships within a triangle. They are defined using a right triangle's structure, with sine representing the ratio of the opposite side to the hypotenuse, \( \sin \theta = \frac{4}{5} \), and cosine representing the ratio of the adjacent side to the hypotenuse, \( \cos \theta = \frac{3}{5} \).
The sum of sine and cosine shows up often in trigonometry problems. For instance, in this problem, we find \( \sin \theta + \cos \theta \) by adding the respective values. These functions are pivotal because they offer a way to model periodic phenomena which include waves and oscillations.

• \( \sin \theta \) tells us how high above the hypotenuse the opposite side rises.
• \( \cos \theta \) describes how far along the hypotenuse the adjacent side stretches.

Understanding sine and cosine helps to unlock complex trigonometric problems and furthers the comprehension of angular relationships.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are the functions derived from the basic trigonometric functions by taking the reciprocal. These include the cosecant (csc), secant (sec), and cotangent (cot) functions. In your exercise, the focus is on \( \csc \theta \) and \( \cot \theta \).
The cosecant function \( \csc \theta \) is the reciprocal of the sine function, which gives \( \csc \theta = \frac{1}{\sin \theta} = \frac{5}{4} \). This calculation provides another perspective on the magnitude of the angle's vertical reach regarding the hypotenuse.
Similarly, the cotangent function \( \cot \theta \) is the reciprocal of the tangent function. Given as \( \cot \theta = \frac{1}{\tan \theta} = \frac{3}{4} \), it offers insight into the inverse of the tangent ratio.

• Reciprocal functions help solve certain trigonometric identities and equations by re-framing the sine, cosine, or tangent in a different light.
• They are especially useful in right triangle scenarios and analytical trigonometric calculations.

These functions are critical when addressing the full scope of trigonometric operations.