Chapter 5: Trigonometric Ratios

5.1 Trigonometric Ratios

Identifying Sides in a Right-Angled Triangle

In a right-angled triangle relative to an acute angle $\theta$:

• Hypotenuse ($H$): The side opposite to the right angle ($90^\circ$). It is always the longest side.
• Opposite Side ($O$): The side directly opposite to the designated angle $\theta$.
• Adjacent Side ($A$): The side adjacent (next) to the angle $\theta$, between $\theta$ and the right angle.

The Three Core Trigonometric Ratios

$$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{O}{H}$$ $$\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{A}{H}$$ $$\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{O}{A}$$

Relationship between Sine, Cosine, and Tangent:

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

Values of Trigonometric Ratios for Special Angles

The table below summarizes the exact values of trigonometric ratios for $30^\circ$, $45^\circ$, and $60^\circ$:

Ratio / Angle ($\theta$)	$30^\circ$	$45^\circ$	$60^\circ$
$\sin \theta$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$
$\cos \theta$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$
$\tan \theta$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$

Trigonometric Ratios of Complementary Angles

Since the two acute angles in a right-angled triangle add up to $90^\circ$ (complementary angles):

$$\sin (90^\circ - \theta) = \cos \theta$$ $$\cos (90^\circ - \theta) = \sin \theta$$

Inverse Trigonometric Functions

To find an unknown angle when two side lengths are known, use inverse trigonometric functions:

$$\theta = \sin^{-1}\left(\frac{O}{H}\right)$$ $$\theta = \cos^{-1}\left(\frac{A}{H}\right)$$ $$\theta = \tan^{-1}\left(\frac{O}{A}\right)$$

Unit Conversion for Angles

Angles can be measured in degrees ($^\circ$) and minutes ($'$), where $1^\circ = 60'$.

• Convert degrees to degrees and minutes: Multiply the decimal part by $60$.
• Convert minutes to decimal degrees: Divide the minutes by $60$.