Previous - Geometry 1: Types and Properties of Angles and Triangles

Trigonometry, which studies the measure of triangles, takes algebra to the next level. Its most well-known features include the Pythagorean Theorem and the sine, cosine, and tangent ratios.

**Pythagoras’ Theorem**

Pythagoras’ theorem states that for a right angled triangle, if a square was to be drawn on each side of the triangle, the area of the square on the hypotenuse side (the side opposite the 90° angle), is equal to the sum of the area of the two squares drawn on the other two sides where, a, b and c are the sides of the triangle.

Pythagoras’ Theorem is most often employed to find the length of an unknown side of a triangle, and as such the form:

a^{2} + b^{2} = c^{2}

**Example:**

Find the length of side c cm.

9cm^{2 }+ 12cm^{2 }= c^{2}

81cm^{2 }+ 144cm^{2 }= 225cm^{2}

c^{2} = 225cm^{2}

c = √225 = 15cm

**Trigonometry - Sides of a Right Angled Triangle **

The order in which the names of the sides are determined is listed below:

- The longest side of the triangle is called the Hypotenuse.
- The side opposite to the angle under consideration, in this case A, is called the Opposite.
- The remaining side is called the Adjacent.

Note: the Opposite side is always the side opposite to the angle under consideration.

The three functions are:

**Sine**

ABC is a right angled triangle

**a** is the symbol for the side opposite angle A

**b** is the symbol for the side opposite angle B

**c** is the symbol for the side opposite angle C

**Example**

Given the knowns, we need to find the unknown. The knowns are: hypotenuse = 7 and opposite = 3.5. Therefore, we will need to find sine which is opposite/hypotenuse.

**Solution**

Sin (x) = Opposite / Hypotenuse = 3.5 / 7 = 0.5. Therefore, the unknown which is the Adjacent which is the length of C is 0.5.

Having now known the length of C, we can now find the respective angle. We can obtain this by solving for this equation which is:

sin(x) = 0.5

Next we can re-arrange that into this:

x = sin^{-1 }(0.5)

On the calculator, just key in 0.5 and use the sin^{-1} button to get the answer:

x = 30°

The difference between sine and sine^{-1 }which is the inverse of sine is that sine gives the ratio of opposite/hypotenuse while sine^{-1 }gives us the angle.

- Sine Function: sin(30°) = 0.5
- Inverse Sine Function: sin
^{-1}(0.5) = 30°

**Tangent**

The ratio called tangent (tan) of an acute angle in a right angled triangle is defined as the ratio between the side opposite the angle and the side adjacent to the angle.

ABC is a right angled triangle

Tan A = a/b

Tan B = b/a

Example

Find the angle A

**Solution**

Tan A= a/b = 9/12 = ¾ = 0.75

We need to use the inverse function for tan, tan^{-1}, to find the angle. This can be obtained by punching 0.75 on your calculator and using the inverse of tan to get:

tan^{-1 }= 37º

**Example**

If we have a = 4 and angle A = 37º

Find the side of b.

tan 37º = 5/b

tan 37º · b = 5

0.75· b = 5

b=6.7

**Cosine**

The cosine (cos) of an acute angle in a right angled triangle is the ratio between the side adjacent to the angle and the hypotenuse of the triangle.

ABC is a right angled triangle

Cos A = b/c

Cos B = a/c

**Example:** Use the cosine function to find the angle A giving your answer to the nearest degree.

cos A = 3/5 = 0.6 gives <A= 53º

Shift cos-1 0.6 = 53º

**Example:** Find the side b.

cos 53º = ^{b}/_{5}

b = Cos 53º · 5

b = 0.6 x 5.

b = 3.