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sin x, cos x and tan x

This page is intended as extra information to complement what is in your package. You should read your package that your teacher gave you before using this page - there are better and more in-depth examples in there.

Examine the following triangle:

Notice where the x (the 'angle of interest') is located. In this triangle, the hypotenuse is 5 cm long, the opposite side (opposite the x) is 3 cm long, and the adjacent side is 4 cm long. We can use these values with sin x, cos x and tan x. The general formula for each of the ratios is shown below:

For this triangle, we could replace the side variables with the appropriate values:

We can use this information to find a the value of the angle x using our calculator! We won't do that yet though. There are three examples that you can reference that represent the three types of questions you are going to see:

FOR ALL OF THE FOLLOWING EXAMPLES MAKE SURE THAT YOUR CALCULATOR IS IN DEGREE MODE.

ALSO NOTE THAT THE RIGHT-ANGLE TRIGONOMETRIC RATIOS ONLY WORK ON RIGHT TRIANGLES.

Example 1: Finding the Length of a Missing Side (Part 1)

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Find the length of the side labelled X in the triangle below:

Solution

Your first thought here might be to use the pythagorean theorem. However, if we try that, you will find that there are too many variables to solve for:

You cannot solve this equation! We need another tool!

Examine the information that is given. (Ignore the 90° angle - it just lets us know that we are working with a right triangle). We are given three pieces of information:

  1. The size of our 'angle of interest'. (35°)
  2. The length of the hypotenuse. (10 cm)
  3. The side we want to find is opposite our angle of interest. We want to find the length of the opposite side.. (X)

The question to ask is: "Which of the three right-triangle trigonometric ratios use an angle, the hypotenuse and the opposite side?" Look at the formulas above. The trig ratio to use for this question is sin x.

It uses the angle (called x in the formula), the opposite side, and the hypotenuse. Fill in what you know:

We need to solve this equation for X. To isolate the X, multiply both sides of the equation by 10. That leaves us with the following equation:

Now it is time to turn to the calculator. (Unless you know the value of sin 35° in your head! I didn't think so! :) ) You may need to check with your teacher about the specific key presses needed to enter this expression into your calculator. Two common calculator keypresses are:

(For D.A.L. Calculators)
OR
(For Non-D.A.L. Calculators)

One of these series of keypresses should yield an answer of:

Therefore, the length of the missing side is 5.74 cm. MAKE SURE that you can use your calculator properly to arrive at this answer. If you CANNOT get this answer on your calculator, ask your teacher for assistance.

Example 2: Finding the Length of a Missing Side (Part 2)

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Find the length of the side labelled X in the triangle below:

Solution

What is different from this example compared to the last example? Let's set up the equation and take a look. In this example, we are given the angle of interest (70°), the opposite side (19m) and the adjacent side (X). Which right-angle trigonometric ratio uses the angle, opp and adj?

We will use tan x to solve this problem. Let's fill in what we know:

The difference between this example and the previous example is that the variable we want to solve for is located in the denominator of the ratio. We are still solving for a side, but our algebra is a little more difficult. There is a trick we can use to solve this! Using cross multiplication we can pull a switcheroo. It looks like the following animation:

One the 'switcheroo' has been performed, we end up with the following statement:

Now we need to enter that fraction into the calculator in order to solve for the value of X. Again, depending on your calculator, there are two different possibilities:

(For D.A.L. Calculators)
OR
(For Non-D.A.L. Calculators)

You should arrive at an answer of:

Which means that the length of the missing side is 6.92 m long.

Example 3: Finding the Size of a Missing Angle

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In the last two examples, we explored how to find the length of a missing side. In this example, you will see how to work your calculator to find a missing angle.

Find the size of angle x in the triangle below:

First we must decide which trig ratio to use. Since we are concerned with an angle of interest (X), the adjacent side (13) and the hypotenuse (17), we will use cos x.

Like the other examples, start with filling in what you know:

The x in this example has already been as isolated as we can make it. Now we have to ask our calculator to find out how big x has to be. Two possible keypresses here are:

(For D.A.L. Calculators)
OR
(For Non-D.A.L. Calculators)

You ONLY use the button on your calculator when you are finding an ANGLE. What you should get for this example is:

This means that the measure of angle x in this example is 40° (rounded to the nearest degree).

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