Trigonometric Ratios

The sides and angles of right triangles always have some relationships between them in common. These constants allow you to figure out the measures of some angles and the length of some sides using less information than may at first seem necessary.

Before learning these relationships you need to be familiar with how these sides (or legs) of a triangle may be referred to on the GED math test when there is a question related to trigonometry.

The hypotenuse is always the hypotenuse, but the two legs of a triangle are named depending on where they are from the angle you are referring to. Both legs will be either termed a “side adjacent” or a “side opposite” depending on where they stand in relation to either angle B or angle C. The right angle, in this case A, is not used to refer to legs in trigonometry.

Look at ∠B in the image. The leg that runs into ∠B is line AB and is ∠B’s adjacent side. Across the triangle from ∠B is AC, and this is the opposite side of ∠B. Likewise, side AB is ∠C’s opposite side and AC is the adjacent side to ∠C.

You can use three trigonometric ratios to help you solve for missing side lengths and angle measurements. The formulas of these three ratios will be provided to you during the test on the formula sheet.

Sine Ratio

The ratio of a triangle depends on which of the two non right angles is being referenced, so every triangle has two sine ratios, two cosine ratios and two tangent ratios.

The sine of ∠B in the triangle above is the side opposite ∠B over the hypotenuse:

The sine of ∠C in the triangle above is the side opposite ∠C over the hypotenuse:

Cosine Ratio

The cosine of ∠B in the triangle above is the line adjacent to ∠B over the hypotenuse.

The cosine of ∠C in the triangle above is the line adjacent to ∠C over the hypotenuse.

Tangent Ratio

The tangent of ∠B in the triangle above is the line opposite to ∠B over the line adjacent to ∠B.

The tangent of ∠C in the triangle above is the line opposite to ∠C over the line adjacent to ∠B.

Applying Trigonometric Ratios

These trigonometric ratios might at first seem pretty useless, however they can be very helpful when you want to find a missing value in a right triangle, which could come up on the GED math test.

When applying trigonometric ratios you will probably need a calculator that has the sin, cos and tan buttons, which the one you will be given during the test does.

Here is an example of how to apply a trigonometric ratio. Let’s say you are shown this image and asked to find the length of the base of the triangle.

To decide if you should use sine, cosine or tangent, think about the information you have and the information you need. You have the measure of one angle (besides the right angle), and the leg adjacent to that angle. You need the measure of the leg opposite that angle. Since your calculation will have to do with the legs of the triangle and not the hypotenuse, you will use tangent. Sine and cosine are for when you have or want to know the length of the hypotenuse. Your equation should look like this:

Now to get tan 50°, just hit 50 then the “tan” button on your calculator: about 1.19

Now solve for b.

The length of the base in the above triangle is 9.52 feet.

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  • Anonymous

    hi, today is the 4th time am retaking ged math. I do not know what am really missing. I passed all other parts but its only math that is given me issues

  • Zaakira

    Hi thanks for all the help.. I have just tried to do the above example, however when I hit “tan50” I get an answer of -0.2719006. Any idea why?

    • Obaid

      When I tried the same in my calculator it told me I was using the wrong format and to hit an first and then to enter the value of the angle. This might be the problem you’re having

      • Obaid

        Sorry, I meant tan, not an.

    • Jake

      Your calculator is in radians. You want it in degrees.

    • Robert

      you must have hit the (tan-1) button

  • Pamela

    Zaakira, there are two different kinds of units that can be used to measure angles (like feet and inches are two different units for measuring length). Degrees is the unit that most people are familiar with, and the other one is called radians. Some calculators can be programmed to interpret angles in units of radians instead of in degrees, and it sounds like that is what happened with your calculator. On my TI-83 calculator, you can change the units by hitting the “mode” button, and then highlighting “Degree” instead of “Radian” on the third row from the top of the screen that displays. If you have a different kind of calculator, you should be able to do something similar to change it. Hope that helps!

  • Brenda

    Is this on the GED? I cannot find the formulas you stated would be on the GED formula sheet for Math regarding sin, cos, and tan. I have looked at several GED Formula Sheets, but none of them have it. Does it need to be committed to memory?

    • Robert

      The sin, cos, and tan formula’s won’t show up on the formula sheet. You are suppose to have the formula’s memorized.

  • Charlene Alexander

    The formula is a squared + b squared = csquared Becareful you have to solve for x if the c is missing

  • sin

    im having trouble understanding all of this.. will this be on the test?? i need to know in order to memorize this

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