Question

There are two possible angles \( B \) between \( 0^{\circ} \) and \( 180^{\circ} \) with this value for sine. Find the two angles, and report them so that \( \angle B_{1} \) is the acute angle. ∠B1= and∠B2= ∠B (round these and all remaining answers to 1 decimal place) Thus, two triangles satisfy the given conditions: triangle A 1​ B 1 c1 and triangle A2 B2 C2

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Answer 1

∠B1 = 30.0 degrees

∠B2 = 150.0 degrees

Explanation

To find the two possible angles
`\( B \)` between
`\( 0^(\circ) \)` and
`\( 180^(\circ) \)` with the given value for sine, follow these steps. First, consider the sine function's periodicity, which occurs every
`\( 360^(\circ) \)`. Hence, for a given sine value, two angles exist within the range of
`\( 0^(\circ) \) to \( 360^(\circ) \)` that satisfy the condition.

To determine the angles, begin by finding the reference angle using the inverse sine function. In this case, the reference angle is
`\( 30.0^(\circ) \)`. As sine is positive in both the first and second quadrants, the acute angle
`\( \angle B_(1) \) is \( 30.0^(\circ) \).` The second angle
`\( \angle B_(2) \)` in the second quadrant can be calculated by subtracting the reference angle from
`\( 180^(\circ) \), resulting in \( 150.0^(\circ) \).`

Therefore, the two angles that satisfy the conditions for the given sine value are
`\( \angle B_(1) = 30.0^(\circ) \)` (the acute angle) and
`\( \angle B_(2) = 150.0^(\circ) \)` in the second quadrant.

Answer 2

To find the two angles with the given value for sine, use the inverse sine function (sin⁻¹) on the given sine value. The acute angle is sin⁻¹(β) and the obtuse angle is 180 - sin⁻¹(β).

Step-by-step explanation:

To find the two angles with the given value for sine, we need to use inverse sine function (sin⁻¹) on the given sine value. Let's assume the given sine value is β. Using the inverse sine function, we can find the acute angle as sin⁻¹(β) and the obtuse angle as 180 - sin⁻¹(β).

For example, if the given sine value is 0.6, then the acute angle β₁ = sin⁻¹(0.6) = 36.9°, and the obtuse angle β₂ = 180 - sin⁻¹(0.6) = 143.1°.

Therefore, the two angles are: β₁ = 36.9° and β₂ = 143.1°.