Question

1. consider this right triangle.

determine whether each equation is correct. mark yes or no for each equation

$\sin ( c ) = \frac { 5 } { 12 }$

$\tan ( c ) = \frac { 12 } { 5 }$

$\cos ( b ) = \frac { 12 } { 13 }$

$\tan ( b ) = \frac { 5 } { 12 }$

$\cos ( c ) = \frac { 13 } { 5 }$

$\sin ( b ) = \frac { 5 } { 13 }$

Explanation:

Step1: Recall trigonometric ratios

For a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)

Step2: Analyze \(\sin(C)\)

For \(\angle C\), opposite side \(AB = 24\), hypotenuse \(BC=26\). So \(\sin(C)=\frac{24}{26}=\frac{12}{13}
eq\frac{5}{12}\)

Step3: Analyze \(\tan(C)\)

For \(\angle C\), opposite \(AB = 24\), adjacent \(AC = 10\). So \(\tan(C)=\frac{24}{10}=\frac{12}{5}\)

Step4: Analyze \(\cos(B)\)

For \(\angle B\), adjacent \(AB = 24\), hypotenuse \(BC = 26\). So \(\cos(B)=\frac{24}{26}=\frac{12}{13}\)

Step5: Analyze \(\tan(B)\)

For \(\angle B\), opposite \(AC = 10\), adjacent \(AB = 24\). So \(\tan(B)=\frac{10}{24}=\frac{5}{12}\)

Step6: Analyze \(\cos(C)\)

For \(\angle C\), adjacent \(AC = 10\), hypotenuse \(BC = 26\). So \(\cos(C)=\frac{10}{26}=\frac{5}{13}
eq\frac{13}{5}\)

Step7: Analyze \(\sin(B)\)

For \(\angle B\), opposite \(AC = 10\), hypotenuse \(BC = 26\). So \(\sin(B)=\frac{10}{26}=\frac{5}{13}\)

Answer:

Equation	Yes	No
\(\tan(C)=\frac{12}{5}\)	✔️
\(\cos(B)=\frac{12}{13}\)	✔️
\(\tan(B)=\frac{5}{12}\)	✔️
\(\cos(C)=\frac{13}{5}\)		✔️
\(\sin(B)=\frac{5}{13}\)	✔️