Question

a right triangle has side lengths 7, 24, and 25 as shown below. use these lengths to find \\(\tan a\\), \\(\sin a\\), and \\(\cos a\\).

Explanation:

Step1: Recall trigonometric ratios

In a right triangle, for an acute angle \( A \):

• \( \tan A=\frac{\text{opposite}}{\text{adjacent}} \)
• \( \sin A=\frac{\text{opposite}}{\text{hypotenuse}} \)
• \( \cos A=\frac{\text{adjacent}}{\text{hypotenuse}} \)

For angle \( A \) in \( \triangle ABC \) (right - angled at \( C \)):

• Opposite side to \( A \): \( BC = 7 \)
• Adjacent side to \( A \): \( AC=24 \)
• Hypotenuse: \( AB = 25 \)

Step2: Calculate \( \tan A \)

Using the formula for tangent, \( \tan A=\frac{\text{opposite}}{\text{adjacent}}=\frac{BC}{AC} \)
Substitute \( BC = 7 \) and \( AC = 24 \), we get \( \tan A=\frac{7}{24} \)

Step3: Calculate \( \sin A \)

Using the formula for sine, \( \sin A=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{BC}{AB} \)
Substitute \( BC = 7 \) and \( AB = 25 \), we get \( \sin A=\frac{7}{25} \)

Step4: Calculate \( \cos A \)

Using the formula for cosine, \( \cos A=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{AC}{AB} \)
Substitute \( AC = 24 \) and \( AB = 25 \), we get \( \cos A=\frac{24}{25} \)

Answer:

\( \tan A=\frac{7}{24} \), \( \sin A=\frac{7}{25} \), \( \cos A=\frac{24}{25} \)