A beautiful gable end dormer
from The Log Connection

Sine, Cosine and Tangent

Objective: When done with this lesson, you will have demonstrated how to

    • to find the sine, cosine, or tangent of angle given the side lengths of a right triangle
    • use the sine, cosine, or tangent to find the missing side length in a right triangle
    • use sine, cosine, or tangent to find an unknown measurement from a physical situation or "real-world context".

Approximate completion time: 3 hours

For our last lesson on proportional reasoning, we will look at three ratios called "sine", "cosine", and "tangent". These three ratios are part of the study of mathematics known as "trigonometry". Let's take a brief look into trigonometry from the perspective of ratios and proportional reasoning.

In the last lesson, you looked at dilations of right triangles as part of your homework. You found ratios of the heights to widths. Those ratios are better known as "tangent" ratios. Let's look at that ratio again.

TANGENT:

The tangent of an angle in a right triangle is the ratio of the length of the leg opposite that angle to the length of the leg adjacent to that angle.

We often write "tan" for tangent. Can you find the tangent button on your calculator?

Consider this gable on a house. It has very beautiful lines called "gable posts". They illustrate well the idea of dilating a right triangle and the resultant similar triangles.

Can you see the right triangles?
Can you see that the angles of the triangles do not change in their measure?
Do you believe the triangles are proportionate in size?

Looking at the triangles more closely, you can see the tangent ratios.

Those ratios are the same because the triangles are similar.

If the angles were different, then the ratios would be different. That is why we tie the notion of tangent to the measure of an angle.

Although we don't have measurements for that house gable in the picture, it appears that the heights of those triangles are the same as their bases. That means that the tangent of x is 1/1. Or written concisely, tan x = 1.

If the roof would have been steeper, then the tan x would have been greater than 1.
If the roof would have been less steep, then the tan x would have been less than 1.
As it is, this roof appears to have an "angle of inclination", x, such that the tan x = 1. Do you know what angle that must be? Can you deduce what angle would make a right triangle have both legs of the same length? I think it's 45. Let's check with a calculator.

First, find a scientific calculator.
(That's a calculator with the buttons "sin", "cos", and "tan".)

  • Online calculators: creativearts , calculator.org, or many others
  • On your computer: Windows "Start" button ? "All Programs" ? "Accessories" ? "Calculator". (Be sure to change "view" to "scientific".)
  • Handheld: Many people prefer to use calculator like the TI-83 or TI-84.

Second, be sure that your settings are in "degrees". We will work in other modes in future courses, yet for now we will stick with degrees.

  • One quick test is to find tan 45. Your calculator either will require entering the "45" or "tan" first. Try it both ways.
    You should get tan 45 = 1.
  • If you are not getting tan 45 = 1, then look for either the "DRG" button or the "mode" button for ways to change to degree mode.

Let's look now at how this tangent ratio can be used.

Stair systems must be built in new homes according to building codes.

If the angle of elevation (stair angle) is to be 34 and the length of each tread (the "run") is to be 10", then how big should each riser (the "rise") be for each step?

Solution:

So the rise for each step should be 6.745 inches or roughly 6 ".

If Bob the Builder wanted to make a stair system with 10.5" treads and 7.5" risers, what angle of elevation would that be?

Solution:

Setting up a picture is always the best first step!

Now the question is, what angle would give a tangent value of .7142857...?
Either you could guess and check with your calculator to find such an angle or you can use a powerful tool that's on your calculator.

Try to find the tan-1 or atan button on your calculator. You likely will need to use the "Shift" or "2nd" or "Inv" button first to get the tan-1 button.

To "undo" the tangent of x, we use the "inverse tangent" button, tan-1.

So to solve tan x = .7142857... we have

x = tan-1(.7142857...)

x = 35.5376...

You can check that tan (35.5376...) really is .7142857...
Not many people would have been able to guess that one.

Therefore the angle of elevation will be about 35 .

Be sure you have the main idea here. Ratios. Ratios. Ratios. Ratios. Ratios.

The tangent of an angle is simply the ratio of two lengths.
Since the tangent value of angle can be found by using a calculator, we can use proportions (two ratios equaling each other) to find an unknown length or unknown angle measure.

We'll practice this notion more after we look at the other two basic trigonometric ratios (sine and cosine).

SINE:

The sine of an angle in a right triangle is the ratio of the length of the leg opposite that angle to the length of the hypotenuse of the triangle.

We often write "sin" for sine. Can you find the sine button on your calculator?

COSINE:

The cosine of an angle in a right triangle is the ratio of the length of the leg adjacent that angle to the length of the hypotenuse of the triangle.

We often write "cos" for cosine. Can you find the cosine button on your calculator?

Many people remember the three basic trigonometric ratios with one of these fun little ditties.
Pick your favorite or make your own. In any case, you'll need to know the three ratios.

Be careful to note the angle of reference. "Opposite" of one angle may be "adjacent" to another.

Now let's finish with a few examples showing sine, cosine, and tangent in action.

Problem
Solution
For the triangle shown find
sin (C),
sin (T),
tan (C),
and tan (T).

sin (C) = 60/100

sin (T) = 80/100

tan (C) = 60/80

tan (T) = 80/60

For the triangle shown find
cos (P),
cos (G),
sin (P),
and tan (G).

cos (P) = 5/13

cos (G) = 12/13

sin (P) = 12/13

tan (G) = 5/12

How far up a wall could a 30 foot ladder reach if the ladder makes a 70 angle with the ground?

A picture is the best place to start!

There are only two side lengths we are using. We must choose the correct trig ratio.

The hypotenuse is 30 long and the unknown height y is opposite the 70 angle. That's sine!

Therefore, the ladder can reach up 28.19 feet.

Find the measure of angle C in this diagram.

Since all three side lengths are known, any of the three trig ratios could be used. Let's choose sine.

sin (C) = 60/100
C = sin-1(60/100)
C = 36.87



Now try your hand at some more ratios using sine, cosine, and tangent by downloading and completing this assignment.