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**Trigonometry** is study of the relationships among angles and sides of triangles, especially right triangles. A right triangle contains one right (90°) angle. The side opposite the right angle is the longest side and is called the hypotenuse.

Any other angle in the right triangle is formed by the hypotenuse and the side called the adjacent side. The third side is opposite to the angle.

The sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse.

**sinθ = (opposite / hypotenuse)**

Each of the following triangles depicts a different scale, but θ is the same.

For a given angle, the ratio depends only on the angle.

For the triangles above

sinθ = 0.60

Notice that units in the ratio cancel. Sine has no units.

The angle can be determined using your calculator and finding the inverse sine. For the above triangles,

sin^{-1}(0.60) = 37°

Sine is useful because it provides the relationship between an angle, its opposite side and the hypotenuse.

For example, determine the length of the hypotenuse in this triangle.

Answer:

sinθ = (opposite / hypotenuse)

sin( 21.8°) = (2.00m / hypotenuse)

hypotenuse = (2.00m) / (0.3714) = 5.39 m

The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse.

**cosθ = (adjacent / hypotenuse)**

Each of the following triangles depicts a different scale, but θ is the same.

For a given angle, the ratio depends only on the angle.

For these triangles

cosθ = 0.60

Notice that the sine of one angle is equal to the cosine of the complementary angle.

Cosine is useful because it provides the relationship between an angle, the side adjacent to the angle and the hypotenuse.

For example, in this triangle, find the length of the adjacent side.

Answer:

cosθ = (adjacent / hypotenuse)

cos(21.8°) = (adjacent / 5.39m)

adjacent = (0.9285)(5.39) = 5.00 m

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.

**tanθ = (opposite / adjacent)**

Each of the following triangles depicts a different scale, but θ is the same.

For a given angle, the ratio depends only on the angle.

For these triangles

tanθ = 1.33

Tangent is useful because it provides the relationship between an angle, the side opposite to the angle and the adjacent side.

For example, in this triangle, find the length y.

Answer:

tanθ = (opposite / adjacent)

tan(35.0°) = (y / 9.00m)

y = (0.7002)(9.00) = 6.30 m

(See bottom of page for answers.)

1.

For each of the triangles labeled 1.a to 1.f, calculate the sine of the indicated angle. Use the inverse sine function of your calculator to determine the angle.

2.

For each of the triangles labeled 1.a to 1.f, calculate the cosine of the indicated angle. Use the inverse cosine function of your calculator to determine the angle.

3.

For each of the triangles labeled 1.a to 1.f, calculate the tangent of the indicated angle. Use the inverse tangent function of your calculator to determine the angle.

4.

Prove that for any right angle triangle the sine of one non-right angle is equal to the cosine of the complementary angle.

5.

Prove that for any right angle triangle, the tangent of either non-right angle is equal to the sine of that angle divided by its cosine.

6. to 14.

Use the triangles labelled 6 to 14.

For each triangle, find the missing side (x, y, or R) or the missing angle (θ).

Selected solutions are printed below.

For solutions to all the problems on this page click here.

4.

In the triangle to the right,

X + Y = 90°

so X and Y are complementary.

sinY = y / hypotenuse

cosX = y / hypotenuse

Therefore, sinY = cosX

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