Problem 24 - Algebra and Trigonometry

Question

Sketch a triangle that has acute angle \(\theta,\) and find the other five trigonometric ratios of \(\theta\) . $$ \csc \theta=\frac{13}{12} $$

Step-by-Step Solution

Answer

The five trigonometric ratios are: \( \sin \theta = \frac{12}{13} \), \( \cos \theta = \frac{5}{13} \), \( \tan \theta = \frac{12}{5} \), \( \sec \theta = \frac{13}{5} \), \( \cot \theta = \frac{5}{12} \).

Step 1: Understand the Reciprocal Relationship

Recall that the cosecant function is the reciprocal of the sine function. Hence, we have: \[ \csc \theta = \frac{1}{\sin \theta} \]Given \( \csc \theta = \frac{13}{12} \), which implies that:\[ \sin \theta = \frac{12}{13} \]

Step 2: Use Pythagoras' Theorem to Find Cosine

For a right triangle with opposite side 12 and hypotenuse 13 (as derived from \( \sin \theta = \frac{12}{13} \)), use Pythagoras' theorem to find the adjacent side \(b\):\[ 12^2 + b^2 = 13^2 \]Calculate \(b\):\[ 144 + b^2 = 169 \]\[ b^2 = 25 \]\[ b = 5 \]Therefore, the cosine of the angle \(\theta\) is:\[ \cos \theta = \frac{5}{13} \]

Step 3: Find the Tangent Ratio

Use the sine and cosine ratios to find the tangent:\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{12}{13}}{\frac{5}{13}} = \frac{12}{5} \]

Step 4: Calculate the Secant Ratio

Secant is the reciprocal of cosine:\[ \sec \theta = \frac{1}{\cos \theta} = \frac{13}{5} \]

Step 5: Determine the Cotangent Ratio

Cotangent is the reciprocal of tangent:\[ \cot \theta = \frac{1}{\tan \theta} = \frac{5}{12} \]

Step 6: List the Trigonometric Ratios

Now summarize all the trigonometric ratios:- \( \sin \theta = \frac{12}{13} \)- \( \cos \theta = \frac{5}{13} \)- \( \tan \theta = \frac{12}{5} \)- \( \csc \theta = \frac{13}{12} \)- \( \sec \theta = \frac{13}{5} \)- \( \cot \theta = \frac{5}{12} \)

Key Concepts

Cosecant

Cosecant is one of the six main trigonometric functions. It's the reciprocal of the sine function. This means that

• \( \csc \( \theta \) = \frac{1}{\sin \theta} \).

From the exercise, we know that \( \csc \theta = \frac{13}{12} \). This tells us that if we take the sine of angle \( \theta \), the ratio of the hypotenuse to the opposite side of our right triangle is 13:12. Elsewhere, to find sine from cosecant, we simply flip this fraction. Hence, \( \sin \theta = \frac{12}{13} \).
Cosecant is particularly useful when working with problems where the sine value is very small since its reciprocal is a larger value, making it easier to handle.

Sine

Sine is a primary trigonometric function associated with the ratio of the opposite side to the hypotenuse in a right triangle.

• The equation for sine is \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).

In our problem, we found \( \sin \theta = \frac{12}{13} \) after using the relation \( \sin \theta = \frac{1}{\csc \theta} \). Knowing the sine ratio helps us understand the relationship between the sides of the triangle: the opposite side is 12, and the hypotenuse is 13.
Grasping sine is essential because it's the foundation for understanding other trigonometric ratios.

Cosine

Cosine is another fundamental trigonometric ratio.

• Cosine relates the adjacent side to the hypotenuse in a right triangle, \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \).

After finding the adjacent side using the Pythagorean theorem, \( b = 5 \), the cosine ratio became \( \cos \theta = \frac{5}{13} \). This demonstrates how the cosine gives a different perspective on the triangle's dimensions.
It complements sine well, and these two ratios are pivotal for many mathematical applications and geometric interpretations.

Tangent

The tangent function is inherently related to sine and cosine.

• It shows the ratio of the opposite side to the adjacent side, \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\text{opposite}}{\text{adjacent}} \).

In this scenario, with \( \sin \theta = \frac{12}{13} \) and \( \cos \theta = \frac{5}{13} \), we divide these values to find \( \tan \theta = \frac{12}{5} \). Tangent gives a useful measure of the steepness of the angle \( \theta \), often crucial in fields like physics and engineering.

Pythagorean Theorem

The Pythagorean theorem is a key mathematical tool crucial for working with right triangles. It's expressed by the formula:

• \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the legs, and \( c \) is the hypotenuse.

In our exercise, we used it to find the missing side of the triangle. Given \( 12^2 + b^2 = 13^2 \), we simplified to find that the adjacent side \( b = 5 \).
This theorem is foundational. It not only allows us to compute unknown sides of triangles but also underpins the derivation of trigonometric functions themselves.