Question

note: triangle may not be drawn to scale. suppose a = 9 and b = 11. find an exact value or give least two decimal places: sin(a) = cos(a) = tan(a) = sec(a) = csc(a) = cot(a) =

Explanation:

Step1: Find hypotenuse \( c \)

Use Pythagorean theorem: \( c = \sqrt{a^2 + b^2} \).
Given \( a = 9 \), \( b = 11 \), so \( c = \sqrt{9^2 + 11^2} = \sqrt{81 + 121} = \sqrt{202} \approx 14.21 \).

Step2: Calculate \( \sin(A) \)

\( \sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c} = \frac{9}{\sqrt{202}} \approx \frac{9}{14.21} \approx 0.63 \).

Step3: Calculate \( \cos(A) \)

\( \cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{c} = \frac{11}{\sqrt{202}} \approx \frac{11}{14.21} \approx 0.77 \).

Step4: Calculate \( \tan(A) \)

\( \tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b} = \frac{9}{11} \approx 0.82 \).

Step5: Calculate \( \sec(A) \)

\( \sec(A) = \frac{1}{\cos(A)} = \frac{\sqrt{202}}{11} \approx \frac{14.21}{11} \approx 1.29 \).

Step6: Calculate \( \csc(A) \)

\( \csc(A) = \frac{1}{\sin(A)} = \frac{\sqrt{202}}{9} \approx \frac{14.21}{9} \approx 1.58 \).

Step7: Calculate \( \cot(A) \)

\( \cot(A) = \frac{1}{\tan(A)} = \frac{b}{a} = \frac{11}{9} \approx 1.22 \).

Answer:

\( \sin(A) \approx 0.63 \)
\( \cos(A) \approx 0.77 \)
\( \tan(A) \approx 0.82 \)
\( \sec(A) \approx 1.29 \)
\( \csc(A) \approx 1.58 \)
\( \cot(A) \approx 1.22 \)