Table of Contents
1. Introduction
The sine, cosine and tangent are functions we use in math to calculate angles and sides of triangles. On your calculator these functions are written in short as “sin”, “cos” and “tan”,
Legend
• = Multiplication
∠ = Angle
≈ = Approximately equal to
- You can only apply sine, cosine and tangent on right triangles.
- Check if your calculator is set to "degrees" or "radians" depending on the solution needed. You can change this setting by pressing the "mode" button on your calculator until you see "Deg" or "Rad". Select the one you need.
2. Revision
Right triangles:
These triangles always have one angle of 90 degrees (the other two angles together are also 90 degrees). The angle of 90 degrees is called a right angle.
Angle sum:
A triangle is a shape with a total of three angles. If you add all of the angles together, you’ll always get a total of 180 degrees, whatever shape the triangle has.
3. What are the sine, cosine and tangent?
With the sine, cosine and tangent we determine the ratio of an angle in a right triangle. A well known example of a right triangle is the triangle with the sides 3, 4 and 5. If you take a different triangle with the same ratio, for example a triangle with sides 6, 8 and 10, than the sine, cosine and tangent of an angle will be the same as in the triangle 3, 4, 5. The sine of an angle is simply said, the ratio between two sides of a triangle that determine the angle.
4. Naming the sides of a triangle
To calculate an angle with sine, cosine or tangent, we’ll need the length of two sides. Each side in a right triangle can be named hypotenuse, adjacent or opposite. Naming the sides is always in relation with the angle you are calculating.
The “hypotenuse”:
The hypotenuse is the longest side in a triangle. This is the side opposite of the right angle. The hypotenuse also lies next to the side you are trying to calculate.
The “opposite side”:
The opposite is the side opposite of the given angle. This is also the only side not touching the angle you are calculating.
The “adjacent side“:
The adjacent side also lies along the angle you are trying to calculate. As we already know one of the sides along the given angle is the hypothenuse, the remaining side then has to be the adjacent side.
5. Tangent of an angle
With tangent you can calculate an angle when you know the opposite and the adjacent sides.
The formula for calculating the tangent of an angle is:
Opposite side / Adjacent side = Tangent of the angle
The number you get when you divide two sides, is called the gradient. The gradient is also called the tangent of the angle. With this number you can calculate the angle in degrees.
To convert the tangent of the angle to degrees, we’ll use the inverse tangent. We do this by pressing the buttons *shift* and *tan* on your calculator. You’ll see that your calculator now shows “tan-1”. If you now type in the tangent of the angle, the calculator will calculate the degree of that angle.
- Other names for “tan-1” are: “inverse tangent”, “arc tan” or “atan”.
6. Sine of an angle
In the same way, you can calculate an angle with sine.
The formula for calculating the sine of an angle is:
Opposite side / Hypotenuse = Sine of the angle
7. Cosine of an angle
In the same way, you can calculate an angle with cosine.
The formula for calculating the cosine of an angle is:
Adjacent side / Hypotenuse = Cosine of the angle
8. Tip
An easy way to remember this, is to think of “SohCahToa”. This is also what I use to remember the formulas of sine, cosine and tangent. “SohCahToa” is an abbreviation of the following:
Soh… Sine: Opposite side / Hypotenuse
…Cah… Cosine: Adjacent side / Hypotenuse
…Toa: Tangent: Opposite side / Adjacent side
9. Calculating sides with tangent
If you only know the length of one side and the degree of an angle, you can calculate the remaining sides. When you only press the *tan* button on your calculator, followed by the angle in degrees, your calculator will calculate the angle back to the tangent of the angle.
If we have a look at the formula for calculating the tangent, we can fill in one of the sides and the tangent of the angle. In the example below the question is: What do we divide by the adjacent side, to get a tangent of 0.75.
Let’s first have a look at a simplified example. What do you have to divide by 2 to get 5? Answer: 10. How do we know this? 5•2 = 10. We can also apply this for calculating sides in triangles. Then we have: Tangent of the angel•adjacent side = opposite side. Just like the example below.
10. Formulas
Sine of an angle:
Opposite side / Hypotenuse = Sine of an angle
Cosine of an angle:
Adjacent side / Hypotenuse = Cosine of an angle
Tangent of an angle:
Opposite side / Adjacent side = Tangent of an angle
