Question

I need help with question A and question b!

Answer 1

In a right-angled triangle with a hypotenuse of 12 and angle A denoted as θ, calculations yield tan(37°) ≈ 1.1314. For cos(θ) = 0.2376, solving yields θ ≈ 18.75°.

Let's use the given information to find the values of the trigonometric ratios.

Given a right-angled triangle ABC with hypotenuse h = 12, and angle A denoted by
`\( \theta \)`, we can use the Pythagorean theorem to find the length of the adjacent side (base), denoted as b. The Pythagorean theorem is given by:

`\[ a^2 + b^2 = h^2 \]`

For this triangle:

`\[ a^2 + b^2 = 9^2 + b^2 = 12^2 \]`

`\[ 81 + b^2 = 144 \]`

`\[ b^2 = 63 \]`

`\[ b = √(63) \]`

Now, we can use these values to find the trigonometric ratios:

a)
`\[ \tan \theta = (a)/(b) = (9)/(√(63)) \approx 1.1314 \]`

b) Given
`\( \cos \theta = 0.2376 \)`, we can use the fact that
`\( \cos \theta = (a)/(h) \):`

`\[ 0.2376 = (√(63))/(12) \cos \theta \]`

Solving for
`\( \theta \):`

`\[ \theta = \cos^(-1) \left( (0.2376 \cdot 12)/(√(63)) \right) \]`

`\[ \theta \approx \cos^(-1)(0.9504) \]`

`\[ \theta \approx 18.75^\circ \]`

Summary:

a)
`\( \tan 37^\circ \approx 1.1314 \)`

b)
`\( \theta \approx 18.75^\circ \)`