Problem 45 - Algebra 2

Question

REVIEW Refer to the figure below. If \(\tan x=\frac{10}{24},\) what are \(\sin x\) and \(\cos x ?\) $$ \begin{array}{l}{F \sin x=\frac{26}{10} \text { and } \cos x=\frac{24}{26}} \\\ {G \sin x=\frac{10}{26} \text { and } \cos x=\frac{24}{26}} \\ {H \sin x=\frac{26}{10} \text { and } \cos x=\frac{26}{24}} \\ {J \sin x=\frac{26}{10} \text { and } \cos x=\frac{24}{26}}\end{array} $$

Step-by-Step Solution

Answer

\( \sin x = \frac{10}{26} \) and \( \cos x = \frac{24}{26} \), which matches option G.

Step 1: Understanding Tangent

We start by recognizing that \( \tan x = \frac{10}{24} \) implies that in a right triangle, the opposite side to angle \( x \) is 10 and the adjacent side is 24.

Step 2: Applying the Pythagorean Theorem

We calculate the hypotenuse \( h \) of the right triangle using the Pythagorean theorem: \( h = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26 \).

Step 3: Finding \( \sin x \)

The sine function is defined as the opposite side over the hypotenuse. Therefore, \( \sin x = \frac{10}{26} \).

Step 4: Finding \( \cos x \)

The cosine function is defined as the adjacent side over the hypotenuse. Thus, \( \cos x = \frac{24}{26} \).

Step 5: Verify with Answer Choices

We compare our calculated values of \( \sin x = \frac{10}{26} \) and \( \cos x = \frac{24}{26} \) with the provided options. The correct match is option G.

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan \), is one of the fundamental trigonometric functions. It is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle. For example, if you have a triangle where \( \tan x = \frac{10}{24} \), it means that the side opposite to the angle \( x \) is 10 units long, and the side adjacent is 24 units long. There are a few key points to remember about the tangent function:

• It can be calculated by dividing the sine of an angle by the cosine of the angle: \( \tan x = \frac{\sin x}{\cos x} \).
• It only applies well in a right-angled triangle setup.
• It is periodic with a period of \( \pi \), which means it repeats its values every \( \pi \) radians.

Understanding the tangent function helps in solving many trigonometric problems, especially when dealing with angles and side lengths.

Sine Function

The sine function, represented as \( \sin \), is another essential trigonometric function. It is defined in the context of a right-angled triangle as the ratio of the length of the opposite side to the length of the hypotenuse.So, when asked to find \( \sin x \) given that \( \tan x = \frac{10}{24} \), you first find the hypotenuse using the Pythagorean theorem. In this case, the hypotenuse \( h \) is calculated as:\[h = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26\]With the hypotenuse known, you can find the sine of the angle \( x \) by:\[\sin x = \frac{10}{26} = \frac{5}{13}\]Some important properties of the sine function include:

• Its value ranges from \(-1\) to \(1\).
• It is an odd function, meaning \( \sin(-x) = -\sin(x) \).
• It has a period of \(2\pi\), meaning its wave pattern repeats every \(2\pi\) radians.

These attributes are essential for both basic trigonometry and more advanced applications.

Cosine Function

The cosine function, or \( \cos \), is closely related to the sine function. It is defined as the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle.In the context of \( \tan x = \frac{10}{24} \), and after calculating the hypotenuse to be 26, the cosine of \( x \) is calculated as follows:\[\cos x = \frac{24}{26} = \frac{12}{13}\]Understanding the cosine function's properties can further elucidate its behavior:

• Like sine, its values also range from \(-1\) to \(1\).
• It is an even function, meaning \( \cos(-x) = \cos(x) \).
• The cosine function is periodic with a period of \(2\pi\), similar to the sine function.

The cosine function often helps in assessing angles and relates closely with sine through the identity \( \sin^2(x) + \cos^2(x) = 1 \). This identity is particularly useful in various trigonometric calculations, ensuring relationships between the angles and the sides of triangles are maintained.