The most important thing that I learned this week in Math 10 was how to find the side lengths and angles of right angle triangles in trigonometry using the ratios sine, cosine, and tangent. This is the most important thing that I have learned in the past week because these ratios are extremely useful tools that you can use in different ways to find many angles or side lengths that you couldn’t find with tools such as the Pythagorean theorem. I chose this topic because prior to this week, I didn’t know many tools or formulas surrounding triangles, and these ratios are essential to solving right angle triangles, and are therefore important to know.

The three ratios use the different sides of the right angle triangle, so the first step to finding the side lengths or angles of a triangle is to label each side. The hypotenuse is the longest side of the triangle and is across from the right angle, the opposite is the side across from the angle that is being used, meaning either the angle we are looking for or the angle we are using to find the ratio. Finally, the adjacent is the remaining side, between the right angle and the other angle used.

 

θ – a Greek symbol called ‘theta’ that is used to represent an unknown angle.

The next step to finding a missing side length or angle is to use the ratios, sine, cosine, or tangent. These are formulas that show the ratio between two of the sides of a right triangle and can therefore be used to find one of the sides used in the ratio given the angle and a different side, or to find an angle given two sides of the triangle. The ratio is equivalent to a percentage of how much one side length of the triangle is of another side length of the triangle.  The sine ratio shows what percentage of length of the hypotenuse is the length of the opposite, cosine does the same except with the adjacent side in place of the opposite, and finally tangent shows what percentage of the length of the adjacent side is the length of the opposite side. For any side length or angle that you want to find, you have to choose a ratio to use, where both sides involved in the ratio are involved in the question.

Ratios:

Sine – sinθ =

Cosine – cosθ =

Tangent – tanθ =

SOHCATOA – Acronym to help with remembering the ratios

S – Sine
O – Opposite
H – Hypotenuse

C – Cosine
A – Adjacent
H – Hypotenuse

T – Tangent
O – Opposite
A – Adjacent

 

Finally, you can use the measurements from your triangle in these ratios and isolate the variable by applying the same operations to either side of the equation.

Example of using one of the ratios to find a side length –

 

SOHCATOA

 

Multiply both sides of the equation by 10 to eliminate the fraction:

 

Multiply sin20 by 10 to solve for x:

 

 

 

Example of using one of the ratios to find an angle –

SOHCATOA

 

To isolate the unknown angle, we cancel out tan by multiplying both sides by .