Some Applications of Trigonometry - Class 10 Maths

1. Line of Sight

The line of sight is the straight line drawn from the eye of an observer to the point in the object viewed by the observer.

Key Idea: Whether looking upward or downward, the line of sight always connects the observer's eye directly to the target point.

The straight line $OP$ joining the observer's eye $O$ to the object $P$.

2. Angle of Elevation

The angle of elevation of an object viewed is the angle formed by the line of sight with the horizontal level when the object is above the horizontal level (i.e., when we raise our head to look at the object).

Formula Reference: If height is $h$ and distance along the ground is $d$, then: $$\tan\theta = \frac{\text{Height of object}}{\text{Distance from foot of object}} = \frac{h}{d}$$

Angle of elevation $\theta = \angle ACB$ formed when looking up at point $A$.

3. Angle of Depression

The angle of depression of an object viewed is the angle formed by the line of sight with the horizontal level when the object is below the horizontal level (i.e., when we lower our head to look at the object).

Alternate Interior Angles Rule: Since the horizontal line through the observer's eye is parallel to the ground level, the angle of depression is always equal to the angle of elevation of the observer as seen from the object: $$\text{Angle of Depression} = \text{Angle of Elevation} = \theta$$

Angle of depression $\theta$ from observer $O$ equals the angle of elevation from object $P$.

4. Sun's Altitude

The altitude of the Sun (or Sun's angle of elevation) is the angle of elevation of the Sun from the tip of the shadow of a vertical object on the ground.

As the Sun rises higher in the sky (altitude $\theta$ increases), the length of the shadow decreases.
When the height of a pole equals the length of its shadow ($h = s$), the Sun's altitude is $\tan\theta = \frac{h}{s} = 1 \implies \theta = 45^\circ$.
When the Sun is on the horizon (sunrise/sunset), $\theta \to 0^\circ$ and shadow length grows indefinitely.

Sun's altitude is the angle $\theta$ where $\tan\theta = \frac{\text{Height}}{\text{Shadow}}$.

5. Three Important Trigonometric Identities

For any acute angle $\theta$ ($0^\circ \le \theta \le 90^\circ$), the fundamental Pythagorean identities are:

Identity 1 $$\sin^2\theta + \cos^2\theta = 1$$

Identity 2 $$1 + \tan^2\theta = \sec^2\theta \quad (\text{or } \sec^2\theta - \tan^2\theta = 1) \quad [0^\circ \le \theta < 90^\circ]$$

Identity 3 $$1 + \cot^2\theta = \csc^2\theta \quad (\text{or } \csc^2\theta - \cot^2\theta = 1) \quad [0^\circ < \theta \le 90^\circ]$$

6. Values of Trigonometric Ratios Table

Standard values for all six trigonometric functions across key angles used in practical height and distance calculations:

Values of Trigonometric Ratios for Standard Angles ($0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$)
Ratio / Angle ($\angle A$)	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$
$\sin A$	$0$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	$1$
$\cos A$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	$0$
$\tan A$	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	Not defined
$\csc A$	Not defined	$2$	$\sqrt{2}$	$\frac{2}{\sqrt{3}}$	$1$
$\sec A$	$1$	$\frac{2}{\sqrt{3}}$	$\sqrt{2}$	$2$	Not defined
$\cot A$	Not defined	$\sqrt{3}$	$1$	$\frac{1}{\sqrt{3}}$	$0$