EMAT 6690

Day 4

Right Triangles ... Pythagorean Theorem ... Trig Ratios ... Constructions

Right Triangles

A right triangle is a triangle with one right angle. The side opposite the right angle is called the hypotenuse. The other two sides are called legs.

In order to construct a right triangle, begin by constructing two lines perpendicular to each other. The legs of the triangles will lie on these two perpendicular lines. The hypotenuse will be the segment connecting the end points (which do not form the right angle) of the two legs.

Pythagorean Theorem

The pythagorean theorem states that in a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse.

Example 1

Find the value of x.

Solution:

Example 2

Is a triangle having sides measuring 18, 34, and 39 a right triangle?

Solution:

In a right triangle the hypotenuse will always be the longest side and the legs will always be the two shorter sides. Therefore, we can label our right triangle as it is shown below.

Now use the pythagorean theorem to test if this is a right triangle.

Since the pythagorean theorem yields a false statement, then this triangle with sides measuring 18, 34, and 39 is not a right triangle.

Trigonometric Ratios

Trigonometry means triangle measurement. A ratio of the lengths of two sides of a right triangle is called a trigonometric ratio. The three most common ratios are sine, cosine, and tangent.

Trionometric ratios are related to the acute angles of a right triangle, not the right angle. The value of a trigonometric ratio depends only on the measure of the angle and not on the size of the triangle. The ratios for the sine of angle A, the cosine of angle A, and the tangent of angle A are shown above.

The sine of angle A is equal to the opposite side (side a) divided by the hypotenuse (side c).

The cosine of angle A is equal to the adjacent side (side b) divided by the hypotenuse (side c).

The tangent of angle A is equal to the opposite side divided by the adjacent side.

To solve problems using trigonometric ratios, a calculator or a trigonometric table must be used to find the value of the trigonometic ratio. For example the sin of a 40 degree angle is .6428 rounded to four decimal places.

Example:

Find the value for x.

Solution:

(the value of x has been rounded to four decimal places)

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