Question 7

Given the trigonometric ratios of two acute angles $A$ and $B$ with $\sin A = \frac{5}{13}$ and $\cos B = \frac{8}{17}$ .

How many of the following four statements are true based on the information above?

\text{(1)} \quad \tan A + \tan B = \frac{19}{20}

\text{(2)} \quad \cos A \sin B + \cos B \sin A = \frac{220}{221}

\text{(3)} \quad \frac{\tan A}{\tan B} = \frac{25}{32}

\text{(4)} \quad \frac{\sin A \cos B}{\sin B \cos A} = \frac{2}{9}

Given $\sin A = \frac{5}{13}$ and $\cos B = \frac{8}{17}$ where $A$ and $B$ are acute angles.

From $\sin A = \frac{5}{13}$ , the opposite side is $5$ and the hypotenuse is $13$ . Then the adjacent side is

\text{Adjacent side} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12

So for angle $A$

\sin A = \frac{5}{13}, \quad \cos A = \frac{12}{13}, \quad \tan A = \frac{5}{12}

Angle A

Visualization of angle

A

with

\sin A = \frac{5}{13}

\cos A = \frac{12}{13}

, and

\tan A = \frac{5}{12}

Sin (22.62°) = 5/13Cos (22.62°) = 12/13Tan (22.62°) = 5/12

22.62°0.39 Radian

From $\cos B = \frac{8}{17}$ , the adjacent side is $8$ and the hypotenuse is $17$ . Then the opposite side is

\text{Opposite side} = \sqrt{17^2 - 8^2} = \sqrt{289 - 64} = \sqrt{225} = 15

So for angle $B$

\sin B = \frac{15}{17}, \quad \cos B = \frac{8}{17}, \quad \tan B = \frac{15}{8}

Angle B

Visualization of angle

B

with

\sin B = \frac{15}{17}

\cos B = \frac{8}{17}

, and

\tan B = \frac{15}{8}

Sin (61.93°) = 15/17Cos (61.93°) = 8/17Tan (61.93°) = 15/8

61.93°1.08 Radian

Statement $\text{(1)}$ : $\tan A + \tan B = \frac{19}{20}$

\tan A + \tan B = \frac{5}{12} + \frac{15}{8}

= \frac{10}{24} + \frac{45}{24}

= \frac{55}{24}

So $\tan A + \tan B = \frac{55}{24}$ , not $\frac{19}{20}$ . Statement $\text{(1)}$ is false.

Statement $\text{(2)}$ : $\cos A \sin B + \cos B \sin A = \frac{220}{221}$

\cos A \sin B + \cos B \sin A = \frac{12}{13} \cdot \frac{15}{17} + \frac{8}{17} \cdot \frac{5}{13}

= \frac{180}{221} + \frac{40}{221}

= \frac{220}{221}

Statement $\text{(2)}$ is true.

Statement $\text{(3)}$ : $\frac{\tan A}{\tan B} = \frac{25}{32}$

\frac{\tan A}{\tan B} = \frac{\frac{5}{12}}{\frac{15}{8}}

= \frac{5}{12} \cdot \frac{8}{15}

= \frac{40}{180}

= \frac{2}{9}

So $\frac{\tan A}{\tan B} = \frac{2}{9}$ , not $\frac{25}{32}$ . Statement $\text{(3)}$ is false.

Statement $\text{(4)}$ : $\frac{\sin A \cos B}{\sin B \cos A} = \frac{2}{9}$

\frac{\sin A \cos B}{\sin B \cos A} = \frac{\sin A}{\cos A} \cdot \frac{\cos B}{\sin B}

= \tan A \cdot \frac{1}{\tan B}

= \frac{\tan A}{\tan B}

= \frac{2}{9}

Statement $\text{(4)}$ is true.

Therefore, the true statements are $\text{(2)}$ and $\text{(4)}$ . The number of true statements is $2$ .

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