Problem 79 - 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 \(\cot \theta+\cos \theta ?\)

Step-by-Step Solution

Answer

The sum of \(\cot \theta + \cos \theta\) is 1.35

Step 1: Find Cotangent

First, find the cotangent of angle theta \(\cot \theta\). Cotangent is the reciprocal of tangent. Given that \( \tan \theta = \frac{4}{3} \), so the cotangent \(\cot \theta\) can be found as: \( \cot \theta = \frac{1}{tan \theta} = \frac{3}{4} \)

Step 2: Find Cosine

Next, identify the cosine of angle theta \(\cos \theta\). This can be derived from the Pythagorean identity, \( \cos^{2} \theta = 1 - \sin^{2} \theta \). However, to find \(\sin^{2} \theta\), use the identity \( \sin^{2} \theta = \tan^{2} \theta /(1 + \tan^{2} \theta)\). Substituting the given value for tangent gives \( \sin^{2} \theta = (4/3)^{2} /(1 + (4/3)^{2}) = 16/25\). Substituting this back into the formula for cosine gives \( \cos^{2} \theta = 1 - 16/25 = 9/25 \). Since sine is positive and the angle theta is between -π/2 and π/2, the cosine is also positive. Therefore, \( \cos \theta = \sqrt{9/25} = 0.6\)

Step 3: Sum of Cotangent and Cosine

Finally, find the sum of cotangent and cosine, which is \( \cot \theta + \cos \theta = \frac{3}{4} + 0.6 = 1.35 \)

Key Concepts

Cotangent

The cotangent function, often abbreviated as "cot," is one of the fundamental trigonometric functions. It is defined as the reciprocal of the tangent function. If you know that \(\tan \theta = \frac{4}{3}\), then \(\cot \theta\) would be \(\frac{1}{\tan \theta} = \frac{3}{4}\).

Understanding cotangent helps in solving trigonometric equations where tangent is known. Just remember, cotangent can help us switch perspectives in right triangles.

• It is important to note that while tangent looks at opposite over adjacent side of a triangle, cotangent reverses that order. Thus, it considers the adjacent side over the opposite.
• Since cotangent is the reciprocal, it doesn't exist for angles where \(\tan \theta\) or tangent is zero, because division by zero is undefined.

This relationship is particularly useful in various mathematical problems, especially under scenarios like you find in trigonometric identities, which form the backbone of trigonometry.

Cosine

Cosine, abbreviated as "cos," is another primary trigonometric function that measures the adjacent side over the hypotenuse in a right triangle. In our exercise, determining \(\cos \theta\) required an understanding of trigonometric identities.

We used the Pythagorean identity: \(\cos^{2} \theta = 1 - \sin^{2} \theta\). Cosine also helps in understanding angles and rotations through identities. A positive cosine indicates that the angle is in either the first or fourth quadrant for certain angle intervals.

• To find \(\cos \theta\) when you have \(\sin \theta\) is simple using Pythagorean identity.
• Remember, cosine is always positive in the interval from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Cosine’s fundamental relationship with sine through \(\sin^{2} + \cos^{2} = 1\) embodies the circle's geometry, placing trigonometry firmly in a visual context.

Tangent

Tangent, or "tan," in trigonometry is a pivotal concept that characterizes \(\tan \theta\) as the ratio of the opposite side to the adjacent side in a right triangle. The tangent function provides insight into angle measurements and is used extensively in both pure and applied mathematics.

In our problem, given \(\tan \theta = \frac{4}{3}\), this told us directly about our angle’s position and proportion. This understanding further enabled finding cotangent effortlessly as its reciprocal value, which was \(\frac{3}{4}\).

• The tangent function is periodic with a period of \(\pi\), meaning it repeats its values every \(\pi\) radians.
• It's crucial to consider the sign of tangent depending on the quadrant, as positive in the first and third quadrants.

Comprehending tangent’s behavior is key for solving many mathematical equations and applications involving angle calculations and wave functions.