Trigonometry 2

The Trigonometry 2 exercise appears under the High school geometry Math Mission, Trigonometry Math Mission and Precalculus Math Mission. The objective of this exercise is to help users find the length of sides of triangles knowing all three trigonometric measures of any angle in the triangle.

Types of Problems

This exercise is similar to the exercise, Trignometry 1.5, just that some of the sides are square root values.

There is one type of problem that shows up in this exercise:

1. Use trigonometric measures to solve for missing information of side lengths - This problem has users to look at the trigonometric measures of any angle in a triangle given one side length and solve for any other side length in the triangle. 

Strategies

1. users can remember the formulas using Sal's "Soh Cah Toa" method in which:
   1. SOH - Sine is equal to Opposite over Hypotenuse
   2. CAH - Cosine is equal to Adjacent over Hypotenuse
   3. TOA - Tangent is equal to Opposite over Adjacent
2. To solve these kind of problems, users can treat the measures like variables.
3. For example, in the picture above users can find out which trigonometric measure is related to $\angle ABC$ and the sides AC and AB. AC is the side opposite to the angle and AB is the hypotenuse, so they can use the sin function as it deals with the side opposite to an angle and the hypotenuse of an triangle. So users can say that $\sin(\angle ABC)={\frac {9}{x}}={\frac {\sqrt {2}}{2}}$. So they get a new equation, ${\frac {9}{x}}={\frac {\sqrt {2}}{2}}$. users can solve for $x$ to get to know what's the measure of the missing side.
4. Knowledge of rationalizing denominators can be useful to solve problems in this exercise. You rationalize denominators to simplify fractions like ${\frac {7}{\sqrt {10}}}$. To simplify this, you multiply both sides of the fraction by $\sqrt{10}$ which gives you ${\frac {\sqrt[{7}]{10}}{10}}$ as $\sqrt{10}$ times $\sqrt{10}$ is 10.

Real-life Applications

1. Trigonometry is used in navigation to find the distance between the shore and a point from the sea.
2. Architects use trigonometry to build towers, etc.
3. Trigonometry is also used in finding the distance between celestial bodies
4. Trigonometry has many applications in astronomy, music, analysis of financial markets, and many more professions.