Question

for each of the triangles below, find: \\(\sin (a), \cos (a), \tan (a), \sec (a), \csc (a), \cot (a)\\) (triangle with right angle, one leg 5, one leg 9, angle a at the top left)

Explanation:

Step1: Find Hypotenuse

Using Pythagorean theorem: \( c = \sqrt{a^2 + b^2} \), where \( a = 5 \), \( b = 9 \).
\( c = \sqrt{5^2 + 9^2} = \sqrt{25 + 81} = \sqrt{106} \)

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

\( \sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{9}{\sqrt{106}} = \frac{9\sqrt{106}}{106} \)

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

\( \cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{5}{\sqrt{106}} = \frac{5\sqrt{106}}{106} \)

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

\( \tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{9}{5} \)

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

\( \sec(A) = \frac{1}{\cos(A)} = \frac{\sqrt{106}}{5} \)

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

\( \csc(A) = \frac{1}{\sin(A)} = \frac{\sqrt{106}}{9} \)

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

\( \cot(A) = \frac{1}{\tan(A)} = \frac{5}{9} \)

Answer:

\( \sin(A) = \boldsymbol{\frac{9\sqrt{106}}{106}} \), \( \cos(A) = \boldsymbol{\frac{5\sqrt{106}}{106}} \), \( \tan(A) = \boldsymbol{\frac{9}{5}} \), \( \sec(A) = \boldsymbol{\frac{\sqrt{106}}{5}} \), \( \csc(A) = \boldsymbol{\frac{\sqrt{106}}{9}} \), \( \cot(A) = \boldsymbol{\frac{5}{9}} \)