Solve each problem. See Examples 3 and 4. Distance through a Tunnel A tunnel is to be built from point A to point B. Both A and B are visible from C. If AC is 1.4923 mi and BC is 1.0837 mi, and if C is 90°, find the measures of angles A and B.

Verified step by step guidance

1. Identify the triangle formed by points A, B, and C, where C is the vertex with a right angle (90°). Since C is 90°, triangle ABC is a right triangle with AC and BC as the legs, and AB as the hypotenuse.
2. Recall that in a right triangle, the sum of the angles is 180°, and since angle C is 90°, the other two angles A and B must add up to 90°.
3. Use the definitions of sine, cosine, or tangent to find the measures of angles A and B. For example, to find angle A, use the tangent function: \(\tan(A) = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{BC}{AC}\).
4. Calculate angle A by taking the inverse tangent (arctan) of the ratio \(\frac{BC}{AC}\): \(A = \arctan\left(\frac{BC}{AC}\right)\).
5. Find angle B by subtracting angle A from 90°: \(B = 90^\circ - A\).

Key Concepts

Right Triangle Properties

A right triangle has one angle measuring 90°, which allows the use of specific trigonometric relationships. In this problem, angle C is 90°, making triangle ABC a right triangle. This simplifies calculations since the sum of the other two angles must be 90°, and the Pythagorean theorem applies.

Trigonometric Ratios (Sine, Cosine, Tangent)

Trigonometric ratios relate the angles of a right triangle to the lengths of its sides. For example, sine of an angle is the ratio of the opposite side to the hypotenuse. Using the given side lengths AC and BC, these ratios help find the unknown angles A and B.

Angle Sum Property of Triangles

The sum of the interior angles in any triangle is always 180°. Since angle C is 90°, angles A and B must add up to 90°. This property allows finding one angle if the other is known, ensuring the solution is consistent.