Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact values of the trigonometric functions for the acute angle \(\theta\). $$\cos \theta=\frac{8}{17}$$

Step-by-Step Solution

Answer

\( \sin \theta = \frac{15}{17} \), \( \tan \theta = \frac{15}{8} \).

Step 1: Identify the Trigonometric Relationship

Given that \( \cos \theta = \frac{8}{17} \), use the identity \( \cos \theta = \frac{adjacent}{hypotenuse} \). This indicates that the side adjacent to angle \( \theta \) is 8 units long, and the hypotenuse is 17 units long.

Step 2: Apply the Pythagorean Theorem

To find the opposite side, use the Pythagorean theorem: \( a^2 + b^2 = c^2 \),where \( a \) is the adjacent side, \( b \) is the opposite side, and \( c \) is the hypotenuse. Substituting the known values, we have:\[ 8^2 + b^2 = 17^2 \]\[ 64 + b^2 = 289 \]\[ b^2 = 225 \]\[ b = 15 \].Thus, the opposite side is 15 units long.

Step 3: Calculate \( \sin \theta \)

The sine of an angle is the quotient of the length of the opposite side and the hypotenuse: \( \sin \theta = \frac{opposite}{hypotenuse} \). Using the values calculated:\[ \sin \theta = \frac{15}{17} \].

Step 4: Calculate \( \tan \theta \)

The tangent of an angle \( \theta \) is given by the ratio of the opposite to the adjacent side: \( \tan \theta = \frac{opposite}{adjacent} \). Using the values found:\[ \tan \theta = \frac{15}{8} \].

Step 5: Verify Using Trigonometric Identities

Verify the values using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):\[ \left( \frac{15}{17} \right)^2 + \left( \frac{8}{17} \right)^2 = 1 \]\[ \frac{225}{289} + \frac{64}{289} = \frac{289}{289} = 1 \].The result holds; thus, the calculations are verified.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that establishes the relationship between the sides of a right-angled triangle. This theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it's expressed as:

• \( a^2 + b^2 = c^2 \)

where \( a \) and \( b \) are the triangle's legs, and \( c \) is the hypotenuse.
Using this theorem helps us find any missing side of a right triangle if two side lengths are known. In our exercise, knowing

The Role in the Problem:

Given: \( \cos \theta = \frac{8}{17} \). The adjacent side is 8, and the hypotenuse is 17. To find the opposite side, apply:

• \( 8^2 + b^2 = 17^2 \)
• Solve: \( 64 + b^2 = 289 \)
• Leads to: \( b^2 = 225 \)
• So, \( b = 15 \)

The opposite side is crucial for determining other trigonometric functions like sine and tangent.

Sine Function

The sine function is fundamental in trigonometry. For a right-angled triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. This is expressed as:

• \( \sin \theta = \frac{opposite}{hypotenuse} \)

This relationship is crucial for understanding how angles relate to side lengths. In our exercise, we calculated:

Applying Sine to the Problem:

With the opposite side found to be 15 and hypotenuse as 17:

• \( \sin \theta = \frac{15}{17} \)

Knowing \( \sin \theta \) is essential for comprehending how angle \( \theta \) directly relates to these side lengths. This ratio tells us how steep or how much incline the triangle has, relative to the angle \( \theta \).

Cosine Function

The cosine function measures the ratio between the adjacent side and the hypotenuse in a right triangle. It's defined by the expression:

• \( \cos \theta = \frac{adjacent}{hypotenuse} \)

This function helps in understanding how the adjacent side scales with the hypotenuse.

Utilizing Cosine in Our Exercise:

We were given \( \cos \theta = \frac{8}{17} \) directly in our initial problem:

• The adjacent side was 8, and the hypotenuse was 17.
• Had this been unknown, we could find these values by rearranging and solving the formula \( \cos \theta = \frac{8}{17} \).

Overall, the cosine function establishes a foundational understanding of the angles' effect on side ratios. Here, it immediately provided us a gateway to explore other aspects of our triangle.

Tangent Function

The tangent function in trigonometry is the ratio of the length of the opposite side to that of the adjacent side in a right triangle. This is expressed with the formula:

• \( \tan \theta = \frac{opposite}{adjacent} \)

Understanding tangent is crucial for gaining insights into how an angle can define the steepness or gradient level between the opposite and adjacent sides of a triangle.

Finding Tangent in Our Solution:

Using the values already found, the tangent function was calculated as:

• \( \tan \theta = \frac{15}{8} \)

This value helps us understand how much of the triangular shape is "tilted" or inclined with respect to the base (adjacent side). It's particularly useful in real-world problems involving slopes, inclines, or whenever such a ratio is required.