Solving Trigonometry Problems: High School Math Help

Trigonometry Help

The Pythagorean Theorem and Similar Triangles

Trigonometry is based on the Pythagorean theorem, which states that, for a right triangle with hypotenuse 'c' and sides 'a' and 'b,' a^2 + b^2 = c^2. If you know the lengths of two of the triangle's sides, you can use this formula to figure out the length of the third side.

Similarity means that if two triangles have the same three angle measures, then their side lengths will be proportional. For example, let's start with the basic 30-60-90 right triangle with sides 1:(radical 3):2. We'll call this Triangle A.

We can create a similar triangle (Triangle B) by multiplying all three sides by the same number. Let's use three to get a new 30-60-90 right triangle with sides 3:3(radical 3):6. We can show that these two triangles are similar by dividing each side of Triangle B by the corresponding side of Triangle A:

3/1 = 3(radical 3)/(radical 3) = 6/2

3 = 3 = 3

In ancient times, mathematicians used the Pythagorean theorem, along with the concept of similar triangles, to calculate the ratios of the side lengths of right triangles with all conceivable angle measures. The good news for modern humans is that we can use these ratios to calculate all angles and side lengths for a triangle if we know the lengths or measures of a total of three sides and/or lengths.

The Ratios

Often, we don't need to figure out the measures of all of a triangle's angles and side lengths. Instead, we just want to find one of these measures, and trigonometric ratios let us do that with very little information.

Sine

One of the basic trigonometric ratios is called the sine (pronounced 'sign' and abbreviated 'sin'). To calculate it, divide the length of the side opposite the angle you want to measure by the length of the triangle's hypotenuse (sin = opp/hyp). Then, use a calculator or trigonometry table to find the angle (called theta) that's associated with that sine ratio.

Cosine

The ratio for cosine (abbreviated 'cos') is found by dividing the side adjacent to theta (the angle you want to find) by the hypotenuse. If you know the lengths of these two sides, you can solve cos = adj/hyp, and then use your calculator or table to find the angle measure (theta) for that ratio.

Tangent

The third major trigonometric ratio is called the tangent (abbreviated 'tan'), and it's equal to the side opposite theta divided by the side adjacent to theta (tan = opp/adj). Once you find the tangent of theta, you can find theta with a calculator or table.