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Math  /  Trigonometry

QuestionNote: Triangle may not be drawn to scale. Suppose c=9\mathrm{c}=9 and A=35\mathrm{A}=35 degrees. Find: a=a= b=b= B=B= □\square degrees

Studdy Solution

STEP 1

1. The triangle is a right triangle.
2. The hypotenuse c=9 c = 9 .
3. Angle A=35∘ A = 35^\circ .

STEP 2

1. Recall the trigonometric relationships in a right triangle.
2. Use the sine function to find side a a .
3. Use the cosine function to find side b b .
4. Use the complementary angle relationship to find angle B B .

STEP 3

Recall the trigonometric relationships for a right triangle:
- sin⁡A=oppositehypotenuse\sin A = \frac{\text{opposite}}{\text{hypotenuse}} - cos⁡A=adjacenthypotenuse\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}

STEP 4

Use the sine function to find side a a :
sin⁡A=ac \sin A = \frac{a}{c} sin⁡35∘=a9 \sin 35^\circ = \frac{a}{9}
Solve for a a :
a=9×sin⁡35∘ a = 9 \times \sin 35^\circ
Calculate a a :
a≈9×0.5736 a \approx 9 \times 0.5736 a≈5.1624 a \approx 5.1624

STEP 5

Use the cosine function to find side b b :
cos⁡A=bc \cos A = \frac{b}{c} cos⁡35∘=b9 \cos 35^\circ = \frac{b}{9}
Solve for b b :
b=9×cos⁡35∘ b = 9 \times \cos 35^\circ
Calculate b b :
b≈9×0.8192 b \approx 9 \times 0.8192 b≈7.3728 b \approx 7.3728

STEP 6

Use the complementary angle relationship to find angle B B :
B=90∘−A B = 90^\circ - A B=90∘−35∘ B = 90^\circ - 35^\circ B=55∘ B = 55^\circ
The lengths of the sides and the measure of angle B B are:
a≈5.1624 a \approx 5.1624 b≈7.3728 b \approx 7.3728 B=55∘ B = 55^\circ

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