Problem 30 - Precalculus

Question

Find the indicated trigonometric function values. If \(\sin \theta=-\frac{7}{25},\) and the terminal side of \(\theta\) lies in quadrant \(1 \mathrm{V},\) find \(\cos \theta\)

Step-by-Step Solution

Answer

\(\cos \theta = \frac{24}{25}\)

Step 1: Understand the Sine Information

We know that \(\sin \theta = -\frac{7}{25}\). In trigonometry, \(\sin \theta\) can be defined as the ratio of the opposite side to the hypotenuse of a right triangle. This information tells us that the opposite side is \(-7\) (since sine is negative in quadrant IV) and the hypotenuse is \(25\).

Step 2: Apply Pythagorean Theorem

To find \(\cos \theta\), we need to find the adjacent side. We'll use the Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse. Substitute \(b = -7\) and \(c = 25\): \(a^2 + (-7)^2 = 25^2\) or \(a^2 + 49 = 625\).

Step 3: Solve for Adjacent Side

Isolate \(a^2\) by subtracting \(49\) from both sides to find the length of the adjacent side.\[a^2 = 625 - 49 = 576\] Taking the square root, \[a = \sqrt{576} = 24\].

Step 4: Determine Cosine

In quadrant IV, cosine is positive. Therefore, cosine is the ratio of the adjacent side to the hypotenuse: \(\cos \theta = \frac{24}{25}\).

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions used in geometry and trigonometry. It relates a right-angled triangle's angle to the ratio of its opposite side over its hypotenuse. When we say \( \sin \theta = -\frac{7}{25} \), this means that in a triangle with angle \( \theta \), the length of the side opposite the angle is \(-7\) and the hypotenuse is \(25\). The negative sign indicates the direction, showing that sine is negative in specific quadrants.

• In quadrant IV, sine values are negative because the opposite side is below the x-axis.
• Understanding sine is crucial because it helps us solve various trigonometric problems by providing the relationship of the sides in a triangle.

Recalling that the sine function can sometimes be counterintuitive in different quadrants, it's essential to recognize these signs to ensure solutions remain correct when calculating angles and triangle properties.

Cosine Function

The cosine function is another critical aspect of trigonometry, often paired with the sine function to solve triangles. Cosine relates the adjacent side and hypotenuse in a right-angled triangle. In mathematical terms, \( \cos \theta = \frac{adjacent}{hypotenuse} \). To find the cosine when given the sine, you might need to first find the length of the adjacent side using the Pythagorean theorem.

• In our exercise, after identifying the opposite side and hypotenuse, the adjacent side is calculated to be \(24\).
• As the cosine is positive in quadrant IV, we find \( \cos \theta = \frac{24}{25} \).

Cosine is fundamental when working out rotations or translating locations in polar coordinates since it helps determine the x-coordinate (or real component) of the position on a plane as an angle varies.

Pythagorean Theorem

The Pythagorean theorem is a foundational principle that applies to right triangles. It states that in a right triangle, the square of the hypotenuse length (\( c \)) is equal to the sum of the squares of the other two sides (\( a \) and \( b \)). The mathematical representation is \( a^2 + b^2 = c^2 \). This rule allows us to find one side's length if the other two are known. In our example:

• We were given the values of \( b = -7 \) and \( c = 25 \).
• Substituting these into the formula, \( a^2 + (-7)^2 = 25^2 \).
• Solving, you find \( a^2 = 576 \), and thus \( a = 24 \).

Understanding the Pythagorean theorem is imperative in mathematics as it forms the bridge between algebra and geometry, fundamental in proving various mathematical concepts and solving complex problems involving distance and length.