Question

1. 4k given the triangle, determine the value of each of the following.
2. ( sin a = )
3. ( cos a = )
4. ( \tan a = )
5. ( sin b = )
6. ( cos b = )
7. ( \tan b = )

(the triangle is a right triangle with right angle at b, ab = 9, bc = 12, ac = 15)

Explanation:

Step1: Recall trigonometric ratios

In a right triangle, the trigonometric ratios are defined as:

• Sine of an angle: $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$
• Cosine of an angle: $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
• Tangent of an angle: $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

For angle \( A \):

• Opposite side to \( A \): \( BC = 12 \)
• Adjacent side to \( A \): \( AB = 9 \)
• Hypotenuse: \( AC = 15 \)

Step2: Calculate \( \sin A \)

\( \sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{12}{15} = \frac{4}{5} \)

Step3: Calculate \( \cos A \)

\( \cos A = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AB}{AC} = \frac{9}{15} = \frac{3}{5} \)

Step4: Calculate \( \tan A \)

\( \tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AB} = \frac{12}{9} = \frac{4}{3} \)

For angle \( B \) (which is a right angle, \( 90^\circ \)):

• \( \sin B = \sin 90^\circ = 1 \)
• \( \cos B = \cos 90^\circ = 0 \)
• \( \tan B \) is undefined (since \( \tan 90^\circ = \frac{\text{opposite}}{\text{adjacent}} \) and adjacent side is 0)

For angle \( C \):

• Opposite side to \( C \): \( AB = 9 \)
• Adjacent side to \( C \): \( BC = 12 \)
• Hypotenuse: \( AC = 15 \)

Step5: Calculate \( \sin C \)

\( \sin C = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{AB}{AC} = \frac{9}{15} = \frac{3}{5} \)

Step6: Calculate \( \cos C \)

\( \cos C = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{12}{15} = \frac{4}{5} \)

Step7: Calculate \( \tan C \)

\( \tan C = \frac{\text{opposite}}{\text{adjacent}} = \frac{AB}{BC} = \frac{9}{12} = \frac{3}{4} \)

Answer:

• \( \sin A = \frac{4}{5} \), \( \cos A = \frac{3}{5} \), \( \tan A = \frac{4}{3} \)
• \( \sin B = 1 \), \( \cos B = 0 \), \( \tan B \) undefined
• \( \sin C = \frac{3}{5} \), \( \cos C = \frac{4}{5} \), \( \tan C = \frac{3}{4} \)