1. Solving Right Triangles In chapter 7, we defined the trigonometric functions in terms of coordinates of points on a circle. Now, our emphasis shifts from circles to triangles. When certain parts (sides and angles) of a triangle are known, you will see that trigonometric relationships can be used to find the unknown parts. This is called solving a triangle. For example, if you know the lengths of the sides of a triangle, then you can find the measures of its angles. In this section, we will consider how trigonometry can be applied to right triangles.

3. Example 1. For the right triangle ABC shown, find the value of b to three significant digits. Which trig ratio should we use to find b? How could we find c?

4. Example 2. The safety instructions for a 20 ft. ladder indicate that the ladder should not be inclined at more than a 70º angle with the ground. Suppose the ladder is leaned against a house at this angle. Find (a) the distance x from the base of the house to the foot of the ladder and (b) the height y reached by the ladder. The foot of the ladder is about 6.84 ft. from the base of the house. The ladder reaches about 18.8 ft above the ground.

6. Example 3. The highest tower in the world is in Toronto, Canada, and is 553 m high. An observer at point A, 100 m from the center of the tower’s base, sights the top of the tower. The angle of elevation is  A. Find the measure of this angle to the nearest tenth of a degree.

7. Because we can divide an isosceles triangle into two congruent right triangles, we can apply trigonometry to isosceles triangles. Example 4. A triangle has sides of lengths 8, 8, and 4. Find the measures of the angles of the triangle to the nearest tenth of a degree.

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