Problem 21 - Algebra and Trigonometry

Question

\(\theta\) is an acute angle and sin u is given. Use the Pythagorean identity \(\sin ^{2} \theta+\cos ^{2} \theta=1\) to find cos \(\theta.\) $$ \sin \theta=\frac{6}{7} $$

Step-by-Step Solution

Answer

By substituting the given sine value in the Pythagorean identity and calculating, we find that \(\cos \theta = \frac{\sqrt{13}}{7}\).

Step 1: Identify given values

The given sine value, \(\sin \theta\), is \(\frac{6}{7}\).

Step 2: Apply Pythagorean identity

Use the Pythagorean trigonometric identity \(\sin^{2}\theta + \cos^{2}\theta = 1\). Substitute the given sine value into the identity equation giving \(\left(\frac{6}{7}\right)^{2} + \cos^{2}\theta = 1\).

Step 3: Solve for cosine theta

Rearrange the equation to isolate \(\cos^{2} \theta\). Then, the resulting equation would be \(\cos^{2} \theta = 1 - \left(\frac{6}{7}\right)^{2}\). After performing the subtraction operation, take the square root on both sides of the equation to find the value for \(\cos \theta\). Note that since \(\theta\) is acute, \(\cos \theta\) will be positive.

Key Concepts

trigonometric identity

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable. They play a crucial role in simplifying trigonometric expressions and solving trigonometric equations. One of the most fundamental trigonometric identities is the Pythagorean identity:

• \[ \sin^2 \theta + \cos^2 \theta = 1 \]

This identity expresses a relationship between the sine and cosine of an angle \( \theta \). It's named after the Pythagorean theorem because of the connection it has with the lengths of a right triangle's sides.
By using this identity, we can calculate one trigonometric function if we know the other. This property is especially helpful with angles in right triangles. Lastly, remember that the Pythagorean identity holds true for any angle, not just acute angles.

sine and cosine relationship

The relationship between sine and cosine is central to understanding trigonometry. Given an angle \( \theta \), the sine function represents the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle. Conversely, the cosine function represents the ratio of the length of the adjacent side to the hypotenuse.
In our example, we know that \( \sin \theta = \frac{6}{7} \). This tells us that the opposite side length is 6 and the hypotenuse is 7 for a right triangle involving this angle. By applying the Pythagorean identity, we can find the cosine of the angle. We substitute the known sine value into the identity:

• \[ \left(\frac{6}{7}\right)^2 + \cos^2 \theta = 1 \]
• This simplifies to: \[ \cos^2 \theta = 1 - \left(\frac{6}{7}\right)^2 \]

This allows us to calculate \( \cos \theta \), knowing that \( \theta \) is acute, which ensures that \( \cos \theta \) is positive.

acute angle properties

An acute angle is an angle that measures less than 90 degrees. These angles are critical in trigonometry because their sine and cosine values are both positive. This property greatly influences how we solve trigonometric equations.
When given that \( \theta \) is an acute angle, we can be certain that both \( \sin \theta \) and \( \cos \theta \) must be positive. This simplifies finding \( \cos \theta \) using the Pythagorean identity because we don't have to concern ourselves with negative values.
In our exercise, since it's specified \( \theta \) is acute and \( \sin \theta = \frac{6}{7} \), accordingly, \( \cos \theta \) is a positive real number. After calculating \( 1 - \left(\frac{6}{7}\right)^2 \), take the positive square root. This reflects the fact that both sine and cosine are positive in the first quadrant where acute angles lie.