Introduction to Trigonometry

Trigonometry (from Greek trigonon "triangle" + metron "measure")

Want to learn Trigonometry? Here is a quick summary.
Follow the links for more, or go to Trigonometry Index

triangle Trigonometry ... is all about triangles.

Trigonometry helps us find angles and distances, and is used a lot in science, engineering, video games, and more!

Right-Angled Triangle

The triangle of most interest is the right-angled triangle. The right angle is shown by the little box in the corner:

triangle showing Opposite, Adjacent and Hypotenuse

Another angle is often labeled θ, and the three sides are then called:

 

Why?

Why is this triangle so important?

Imagine we can measure along and up but want to know the direct distance and angle:

triangle showing Opposite, Adjacent and Hypotenuse

Trigonometry can find that missing angle and distance.

Or maybe we have a distance and angle and need to "plot the dot" along and up:

triangle showing Opposite, Adjacent and Hypotenuse

Questions like these are common in engineering, computer animation and more.

And trigonometry gives the answers!

Sine, Cosine and Tangent

The main functions in trigonometry are Sine, Cosine and Tangent

They are simply one side of a right-angled triangle divided by another.

For any angle "θ":

sin=opposite/hypotenuse cos=adjacent/hypotenuse tan=opposite/adjacent

(Sine, Cosine and Tangent are often abbreviated to sin, cos and tan.)

 

Example: What is the sine of 35°?

triangle 2.8 4.0 4.9 has 35 degree angle

Using this triangle (lengths are only to one decimal place):

sin(35°) = OppositeHypotenuse = 2.84.9 = 0.57...

The triangle could be larger, smaller or turned around, but that angle will always have that ratio.

Calculators have sin, cos and tan to help us, so let's see how to use them:

 

right angle triangle 45 degrees, hypotenuse 20

Example: How Tall is The Tree?

We have measured the distance (using a laser) and the angle (using a protractor), so:

Sine is the ratio of Opposite / Hypotenuse:

sin(45°) = Opposite Hypotenuse

calculator-sin-cos-tan

Get a calculator, type in "45", then the "sin" key:

sin(45°) = 0.7071...

0.7071... is the ratio of the side lengths: in other words the Opposite is about 0.7071 times as long as the Hypotenuse.

Let's use that value:

0.7071... = Opposite Hypotenuse

And we also know the hypotenuse is 20:

0.7071... = Opposite 20

Multiply both sides by 20:

20 × 0.7071... = Opposite

Finally:

Opposite = 14.14m (to decimals)

When you gain more experience you can do it quickly like this:

 

right angle triangle 45 degrees, hypotenuse 20

Example: How Tall is The Tree?

Start with:sin(45°) = Opposite Hypotenuse
We know:0.7071... = Opposite 20
Swap sides: Opposite 20 = 0.7071...
Multiply both sides by 20: Opposite = 0.7071... × 20
Calculate:Opposite = 14.14 (to 2 decimals)

The tree is 14.14m tall

Try Sin Cos and Tan

Play with this for a while (move the mouse around) and get familiar with values of sine, cosine and tangent for different angles, such as 0°, 30°, 45°, 60° and 90°.

Also try 120°, 135°, 180°, 240°, 270° etc, and notice that positions can be positive or negative by the rules of Cartesian coordinates, so the sine, cosine and tangent change between positive and negative also.

So trigonometry is also about circles!

unit circle

Unit Circle

What you just played with is the Unit Circle.

It is a circle with a radius of 1 with its center at 0.

Because the radius is 1, we can directly measure sine, cosine and tangent.

Here we see the sine function being made by the unit circle:

Note: you can see the nice graphs made by sine, cosine and tangent.

Degrees and Radians

Angles can be in Degrees or Radians. Here are some examples:

Angle Degrees Radians
right angleRight Angle  90° π/2
__ Straight Angle 180° π
right angle Full Rotation 360° 2π

Repeating Pattern

Because the angle is rotating around and around the circle the Sine, Cosine and Tangent functions repeat once every full rotation (see Amplitude, Period, Phase Shift and Frequency).

When we want to calculate the function for an angle larger than a full rotation of 360° (2π radians) we subtract as many full rotations as needed to bring it back below 360° (2π radians):

Example: what is the cosine of 370°?

370° is greater than 360° so let us subtract 360°

370° − 360° = 10°

cos(370°) = cos(10°) = 0.985 (to 3 decimal places)

And when the angle is less than zero, just add full rotations.

Example: what is the sine of −3 radians?

−3 is less than 0 so let us add 2π radians

−3 + 2π = −3 + 6.283... = 3.283... radians

sin(−3) = sin(3.283...) = −0.141 (to 3 decimal places)

Solving Triangles

Trigonometry is also useful for general triangles, not just right-angled ones .

It helps us in Solving Triangles. "Solving" means finding missing sides and angles.

Example: Find the Missing Angle "C"

trig ASA example

Angle C can be found using angles of a triangle add to 180°:

So C = 180° − 76° − 34° = 70°

We can also find missing side lengths. The general rule is:

When we know any 3 of the sides or angles we can find the other 3
(except for the three angles case)

See Solving Triangles for more details.

Other Functions (Cotangent, Secant, Cosecant)

Similar to Sine, Cosine and Tangent, there are three other trigonometric functions which are made by dividing one side by another:

triangle showing Opposite, Adjacent and Hypotenuse

Cosecant Function:
csc(θ) = Hypotenuse / Opposite
Secant Function:
sec(θ) = Hypotenuse / Adjacent
Cotangent Function:
cot(θ) = Adjacent / Opposite

 

Trigonometric and Triangle Identities

And as you get better at Trigonometry you can learn these:

right angled triangle

The Trigonometric Identities are equations that are true for all right-angled triangles.

triangle

The Triangle Identities are equations that are true for all triangles (they don't have to have a right angle).

 

Enjoy becoming a triangle (and circle) expert!