Triangles, radians and trigonometric graphs

What you’ll learn

• Solve triangle problems with sine and cosine rules.
• Use radian measure for arcs and sectors.
• Recognise trigonometric periods and exact values.

Understand the idea

Choose the cosine rule when two sides and their included angle are known, or when all three sides are known. The sine rule links a side to its opposite angle. With two sides and a non-included angle, a second triangle may exist because supplementary angles have the same sine.

One radian subtends an arc equal in length to the radius. This makes arc and sector calculations compact, but their standard formulae require radians. Convert degrees by multiplying by $\pi/180$. A segment area is the sector area minus the triangle area.

Sine and cosine repeat every $2\pi$ radians, while tangent repeats every $\pi$. The multiplier inside the angle changes the period: $\sin(kx)$ completes a cycle in $2\pi/|k|$. The multiplier outside changes amplitude, and a vertical shift changes the midline.

Key formulae

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

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

$$\text{triangle area}=\frac12ab\sin C$$

$$s=r\theta,\qquad \text{sector area}=\frac12r^2\theta\quad(\theta\text{ in radians})$$

Worked example

A circle has radius $8$ cm. Find the arc length and sector area for an angle of $75^\circ$.

1. Convert to radians: $75\pi/180=5\pi/12$.
2. Arc length is $8(5\pi/12)=10\pi/3$.
3. Sector area is $\frac12(8^2)(5\pi/12)=40\pi/3$.

Answer: Arc length $\frac{10\pi}{3}$ cm; sector area $\frac{40\pi}{3}$ cm$^2$.