Problem 49 - Algebra 2

Question

If the secant of angle \(\theta\) is \(\frac{25}{7}\) , what is the sine of angle \(\theta ?\) A. \(\frac{5}{25}\) B. \(\frac{7}{25}\) C. \(\frac{24}{25}\) D. \(\frac{25}{7}\)

Step-by-Step Solution

Answer

The sine of angle \( \theta \) is \( \frac{24}{25} \).

Step 1: Understand the Relationship

In trigonometry, the secant of angle \( \theta \) is defined as the reciprocal of the cosine of \( \theta \). If \( \sec \theta = \frac{25}{7} \), then \( \cos \theta = \frac{7}{25} \).

Step 2: Use the Pythagorean Identity

The Pythagorean identity states that for an angle \( \theta \), \( \sin^2 \theta + \cos^2 \theta = 1 \). Now that we know \( \cos \theta = \frac{7}{25} \), we can substitute this into the identity to find \( \sin^2 \theta \):\[\sin^2 \theta + \left( \frac{7}{25} \right)^2 = 1\]

Step 3: Solve for \( \sin^2 \theta \)

Substitute the value of \( \cos \theta \) into the identity:\[\sin^2 \theta + \frac{49}{625} = 1\]Subtract \( \frac{49}{625} \) from both sides:\[\sin^2 \theta = 1 - \frac{49}{625} = \frac{625}{625} - \frac{49}{625} = \frac{576}{625}\]

Step 4: Find \( \sin \theta \)

To find \( \sin \theta \), take the square root of both sides:\[\sin \theta = \sqrt{\frac{576}{625}}\]Simplify the square root:\[\sin \theta = \frac{24}{25}\]

Step 5: Confirm the Correct Answer

Check the answer choices for the value of \( \sin \theta \). Option C is \( \frac{24}{25} \), which matches our calculation.

Key Concepts

Secant

The secant of an angle is a concept in trigonometry that might seem a bit mysterious at first, but it's quite straightforward once you get a hang of it. The secant function is the reciprocal of the cosine function. This means that for any angle \( \theta \), the secant is defined as:
\[ \sec \theta = \frac{1}{\cos \theta} \]
This is very useful because if you know the secant of the angle, you can easily find the cosine by flipping the fraction.

• If \( \sec \theta = \frac{25}{7} \), then \( \cos \theta = \frac{7}{25} \).
• Think of secant as always greater than 1 (or less than -1), because cosine's max magnitude is 1.

Understanding secant can help unlock complex trigonometric problems, making it a valuable tool in your math toolkit.

Sine

The sine function is one of the fundamental trigonometric functions, often associated with the opposite side of a right triangle relative to a given angle. It's defined as:
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \]
In many trigonometric problems, including our original exercise, finding the sine involves using other known values and identities.

• When using the Pythagorean identity, you can deduce \( \sin \theta \) if you know \( \cos \theta \).
• Our problem required solving for \( \sin \theta \) once we knew \( \cos \theta = \frac{7}{25} \).
• Once calculated, \( \sin \theta = \frac{24}{25} \), a common trap might be trying to guess rather than solve.

Always remember to check solutions to ensure consistency with known values and identities, securing accurate results.

Pythagorean identity

The Pythagorean identity is one of the most important identities in trigonometry. It ties together the sine and cosine functions and is expressed as:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
This identity is derived from the Pythagorean theorem and applies to any angle. It's very useful for finding the sine when you know the cosine, or vice versa.

• In our exercise, we used the given \( \cos \theta = \frac{7}{25} \) to find \( \sin \theta \).
• After rearranging the identity and solving, we found \( \sin^2 \theta = \frac{576}{625} \).
• Taking the square root, we concluded \( \sin \theta = \frac{24}{25} \), consistent with the identity.

Mastering the Pythagorean identity allows for confident tackling of trigonometric equations and enhances problem-solving versatility.