Problem 18 - Algebra and Trigonometry

Question

\(\theta\) is an acute angle and sin u and cos u are given. Use identities to find tan \(\theta\), csc \(\theta\), sec \(\theta\), and cot \(\theta\). Where necessary, rationalize denominators. $$ \sin \theta=\frac{3}{5}, \quad \cos \theta=\frac{4}{5} $$

Step-by-Step Solution

Answer

The values for the trigonometric identities are: \(tan θ = \frac{3}{4}\), \(csc θ = \frac{5}{3}\), \(sec θ = \frac{5}{4}\), \(cot θ = \frac{4}{3}\).

Step 1: Find tan θ

Use the identity \(tan θ = \frac{sin θ}{cos θ}\). Here, \(sin θ = \frac{3}{5}\) and \(cos θ = \frac{4}{5}\). Substituting these values into the identity, \(tan θ = \frac{3/5}{4/5} = \frac{3}{4}\).

Step 2: Find csc θ

Utilize the identity \(csc θ = \frac{1}{sin θ}\). Substituting \(sin θ = \frac{3}{5}\) into the formula, \(csc θ = \frac{1}{3/5} = \frac{5}{3}\).

Step 3: Find sec θ

Use the identity \(sec θ = \frac{1}{cos θ}\). Substituting \(cos θ = \frac{4}{5}\) into the formula gives \(sec θ = \frac{1}{4/5} = \frac{5}{4}\).

Step 4: Find cot θ

By using the identity \(cot θ = \frac{1}{tan θ}\) we can substitute \(tan θ = \frac{3}{4}\) into the formula. \(cot θ = \frac{1}{3/4} = \frac{4}{3}\).

Key Concepts

Understanding tan \theta (Tangent of Theta)

The tangent of an angle in a right triangle is a fundamental trigonometric function that relates the opposite side to the adjacent side. To understand the concept of tangent, consider a right triangle where tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}.

For our given exercise, we calculate tan \theta using the provided sine and cosine values, which are themselves ratios of the triangle's sides. Remember the identity \(tan \theta = \frac{sin \theta}{cos \theta}\), which is essentially the ratio of sine to cosine. Hence, with \(sin \theta = \frac{3}{5}\) and \(cos \theta = \frac{4}{5}\), the tangent is \(tan \theta = \frac{3}{5} \/ \frac{4}{5} = \frac{3}{4}\), simplifying the complex fraction.

Grasping csc \theta (Cosecant of Theta)

The cosecant function, denoted as csc \theta, is the reciprocal of the sine function. It's less common in basic trigonometry but equally important. The function is defined as \(csc \theta = \frac{1}{sin \theta}\).

Understanding this concept requires recognizing that reciprocals invert the value of a fraction. For the exercise's acute angle \( \theta \), we found csc by taking the reciprocal of \(sin \theta\). Substituting the given value \(sin \theta = \frac{3}{5}\), we get \(csc \theta = \frac{1}{\frac{3}{5}} = \frac{5}{3}\), inverting the sine ratio to find the cosecant.

Deciphering sec \theta (Secant of Theta)

Similar to cosecant, the secant function, sec \theta, is the reciprocal of the cosine function. It's expressed as \(sec \theta = \frac{1}{cos \theta}\).

For students to understand secant, it's essential to grasp the concept of reciprocity, which means flipping a fraction. Upon knowing that \(cos \theta = \frac{4}{5}\) as given in the exercise, we applied the reciprocal identity to find \(sec \theta = \frac{1}{4/5} = \frac{5}{4}\). So, the secant is simply the flipped version of the cosine ratio.

Interpreting cot \theta (Cotangent of Theta)

The cotangent is another fundamental trigonometric function often denoted as cot \theta. It can be considered as the complement to the tangent function, given by the reciprocal identity \(cot \theta = \frac{1}{tan \theta}\).

In our exercise, understanding the cotangent involves recognizing it as the flip of the tangent ratio. Our earlier calculation of \(tan \theta = \frac{3}{4}\) allows us to find the cotangent by inverting this ratio, resulting in \(cot \theta = \frac{1}{\frac{3}{4}} = \frac{4}{3}\). This provides insight into the cotangent as being the adjacent side over the opposite side in a right-angled triangle.