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Math  /  Trigonometry

Question7. For each triangle, determine the length of the hypotenuse to the nearest tenth of a metre. a)

Studdy Solution

STEP 1

1. Each triangle is a right triangle.
2. The given side is the side adjacent to the given angle.
3. We need to find the hypotenuse h h for each triangle.

STEP 2

1. Use the cosine function to find the hypotenuse for triangle a).
2. Use the cosine function to find the hypotenuse for triangle b).

STEP 3

For triangle a), recall the cosine function in a right triangle:
cos⁡(θ)=Adjacent SideHypotenuse \cos(\theta) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}

STEP 4

Substitute the given values for triangle a):
cos⁡(35∘)=3.0 mh \cos(35^\circ) = \frac{3.0 \, \text{m}}{h}

STEP 5

Solve for the hypotenuse h h in triangle a):
h=3.0 mcos⁡(35∘) h = \frac{3.0 \, \text{m}}{\cos(35^\circ)}

STEP 6

Calculate the value of h h for triangle a) using a calculator:
h≈3.00.8192≈3.7 m h \approx \frac{3.0}{0.8192} \approx 3.7 \, \text{m}

STEP 7

For triangle b), recall the cosine function in a right triangle:
cos⁡(θ)=Adjacent SideHypotenuse \cos(\theta) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}

STEP 8

Substitute the given values for triangle b):
cos⁡(39∘)=5.0 mh \cos(39^\circ) = \frac{5.0 \, \text{m}}{h}

STEP 9

Solve for the hypotenuse h h in triangle b):
h=5.0 mcos⁡(39∘) h = \frac{5.0 \, \text{m}}{\cos(39^\circ)}

STEP 10

Calculate the value of h h for triangle b) using a calculator:
h≈5.00.7771≈6.4 m h \approx \frac{5.0}{0.7771} \approx 6.4 \, \text{m}
The lengths of the hypotenuses are:
a) 3.7 m \boxed{3.7 \, \text{m}}
b) 6.4 m \boxed{6.4 \, \text{m}}

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