SL 3.2—2d and 3d trig Notes

Trigonometric Ratios in Right-Angled Triangles

Trigonometric ratios are ratios of sides of right-angled triangles, dependent on the angles in the triangle. To make use of these, we label the sides on a triangle with relation to an angle $\theta$:

1. The hypotenuse is the longest side of the right-angled triangle, opposite the right angle.
2. The opposite side to an angle $\theta$ is the only side the angle does not touch.
3. The adjacent side to an angle $\theta$ is the only non-hypotenuse side the angle touches.

The three primary ratios are sine, cosine, and tangent.

For a right-angled triangle with an angle θ:

• $\sin θ = \frac{\text{opposite}}{\text{hypotenuse}}$
• $\cos θ = \frac{\text{adjacent}}{\text{hypotenuse}}$
• $\tan θ = \frac{\text{opposite}}{\text{adjacent}}$

Additionally, we can express $\tan\theta$ in terms of $\sin\theta$ and $\cos\theta$:

$$\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$$

$$\tan\theta = \frac{\frac{\text{opposite}}{\text{hypotenuse}}}{\frac{\text{adjacent}}{\text{hypotenuse}}}$$

$$\tan\theta = \frac{\sin\theta}{\cos\theta}$$

The Sine Rule

The sine rule is used in non-right angled triangles when we know:

• Two angles and one side, or
• Two sides and the angle opposite one of them

The sine rule states:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Where a, b, and c are the lengths of the sides opposite to angles A, B, and C respectively.

Example

In the following triangle, find $\theta$.

Solution:

$\frac{\sin\theta}{6} = \frac{\sin48°}{11}$

$\theta = \arcsin\frac{6\sin48º}{11} \approx 23.9°$

The Cosine Rule

The cosine rule is used when we know:

• Three sides of a triangle, or
• Two sides and the included angle

The cosine rule states:

$c^2 = a^2 + b^2 - 2ab \cos C$

This can be rearranged to find an angle:

$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$

Example

Find the side length $x$ in the following triangle.

Solution:

$x^2 = 13^2 + 7^2 - 2(7)(13)\cos40°$

$x = \sqrt{13^2 + 7^2 - 2(7)(13)\cos40°} \approx 8.86\ cm$

Area of a Triangle

For any triangle, the area can be calculated using the formula:

$\text{Area} = \frac{1}{2}ab \sin C$

Where a and b are any two sides, and C is the angle between them.

Example

Find the area of the following triangle:

Solution:

$\text{Area} = (20)(13)\sin78°$

$\text{Area} \approx 254\ c=mm^2$