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Math SATs- Trigonometry

Welcome to a Math SAT Preparation Lesson. In this lesson we'll go over some trigonometry, which we'll refer to by its pet name trig.

Before we start with trig, let's go over angles

Angles are found between two lines. They're usually measured in degrees, but they can be measured in radians too. A line is always, always 180 degrees, so if I have a line and another line going through it, the two angles created add up to 180. Also, perpendicular lines are separated by a right-angle, which always measure 90 degrees.

And now, trig. Trig is the mathematical area that deals exclusively with planar triangles.

Recall from the Geometry lesson that triangles are polygons with 3 sides. Also recall the important fact that the sum of the angles in a triangle is always 180 degrees.

There are different types of triangles in our world, and here's a list of the important ones:

1. Equilateral- all sides are equal. As an immediate result, all angles are also equal and are always 60 degrees. These ones are easy and fun.

2. Isosceles- 2 of the sides are equal. The angles opposite of the equal sides have to be equal too.

3. Right angled- one of the angles measure 90 degrees.

There are a lot of neat things about right-angled triangles.

We can use things called the tangent, sine and cosine to measure its angles and/or sides. The thing to remember is SOHCAHTOA. No, this isn't a mumble in mock-Swedish, it's a way of remembering how to measure right-angled triangles. It works like this: Say I have an angle A, which isn't the right-angle of the triangle. Then:

SOH:

CAH:

TOA:

Where the adjacent represents the length of the side adjacent to the angle A and opposite is the length of side opposite to it.

An additional basic trick is the Pythagorean theorem:

The square of the hypotenuse equals the sum of the squares of the other two sides. Together with SOHCAHTOA, this will enable you to calculate all sides and angles of a given right-angled triangle.

But about about non-right-angled triangles? Do we have tricks to deal with them?

Yes, we do, but they're not as clean. The first thing to know is the Sine-Law. Say I got me a triangle with points I call A, B and C, and the sides opposite of each point are respectively a, b and c. Then . This formula will help figure out the sides and angles of any triangle, right-angled or not. It's useful if you're given either 2 sides and one angle opposite of that side, or 2 angles and one side of a triangle and you're asked to find the missing pieces of information.

If you're given 3 sides, or 2 sides and the angle between them, you want to use the Cosine-Law: . The two laws will help figure out the entire set of lengths and angles of any triangle given 3 pieces of information, unless you're given only the 3 angles, in which case there's an infinite amount of different triangles that fit.

It's important to be able to manipulate the formulas to find different variables. In the cos-law for example you should be able to find angle C if you're given all 3 sides by rearranging the equation to look like this : .

Trig looks scary, but once you've had enough practice you'll be able to figure out triangle measurements in your sleep. Go practice your stuff with the trig test.

Image credit : arenamontanus

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