Overview

Trigonometry primarily deals with the study of triangles, specifically right-angled triangles. The foundational elements include trigonometric ratios such as sine, cosine, and tangent, which relate the angles of a triangle to the lengths of its sides.

In the context of right-angled triangles:

• Sine (sin) of an angle is the ratio of the length of the opposite side to the hypotenuse.
• Cosine (cos) of an angle is the ratio of the length of the adjacent side to the hypotenuse.
• Tangent (tan) of an angle is the ratio of the length of the opposite side to the adjacent side.

These ratios are critical in solving various problems involving right-angled triangles and have applications in fields such as physics, engineering, and computer graphics.

Applications

Trigonometry is not just a theoretical subject; it has practical applications in various fields:

• Architecture and Engineering: For designing buildings, bridges, and other structures.
• Astronomy: For calculating distances between celestial bodies.
• Navigation: For determining directions and distances in navigation and aviation.
• Physics: For understanding wave functions, oscillations, and many other phenomena.
• Computer Graphics: For rendering 2D and 3D graphics in video games and simulations.
• Geography: For measuring distances and plotting courses on maps.

Overview

Right-angled triangles are fundamental in the study of trigonometry. A right-angled triangle has one angle that is exactly 90 degrees. The side opposite the right angle is known as the hypotenuse, which is the longest side of the triangle. The other two sides are referred to as the adjacent side (next to the angle of interest) and the opposite side (opposite to the angle of interest).

Trigonometric Ratios

Trigonometric ratios are derived from the sides of a right-angled triangle and are essential for solving problems involving these triangles. The primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). Each of these ratios is a function of an angle and can be defined as follows:

• Sine (sin): The ratio of the length of the opposite side to the hypotenuse.

\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)

• Cosine (cos): The ratio of the length of the adjacent side to the hypotenuse.

\( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)

• Tangent (tan): The ratio of the length of the opposite side to the adjacent side.

\( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

These ratios help in determining the lengths of the sides of a triangle when one side length and one angle (other than the right angle) are known. They also assist in finding the measure of an angle when the lengths of two sides of the triangle are known.

Example

Consider a right-angled triangle where the angle of interest is \( \theta \), the length of the opposite side is 3 units, the length of the adjacent side is 4 units, and the hypotenuse is 5 units.

• Calculating Sine:

\( \sin(\theta) = \frac{3}{5} = 0.6 \)

• Calculating Cosine:

\( \cos(\theta) = \frac{4}{5} = 0.8 \)

• Calculating Tangent:

\( \tan(\theta) = \frac{3}{4} = 0.75 \)

To find a missing length in a right-angled triangle, follow these steps:

1. Identify the given information and what you need to find:

   Determine which sides and angles are known.

   Choose the appropriate trigonometric ratio based on the given information.

2. Set up the equation using the trigonometric ratio:

   If you know an angle and the length of the hypotenuse, use sine or cosine.

   If you know an angle and the length of one side, use tangent.

3. Solve for the missing length:

   Rearrange the equation to isolate the unknown side.

   Use a calculator to compute the value if necessary.

How to Find Missing Angles

To find a missing angle in a right-angled triangle, follow these steps:

1. Identify the given information and what you need to find:

   Determine which sides are known.

2. Set up the equation using the inverse trigonometric function:

   Use the appropriate ratio (sin, cos, or tan) based on the known sides.

3. Solve for the missing angle:

   Use a calculator to find the inverse trigonometric function value.

Using Special Angles

Certain angles, known as special angles, have well-known sine, cosine, and tangent values that can be used to simplify calculations. These angles include \(30^\circ\), \(45^\circ\), and \(60^\circ\).

• For \(30^\circ\):

  \( \sin(30^\circ) = \frac{1}{2} \)

  \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \)

  \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \)

• For \(45^\circ\):

  \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \)

  \( \cos(45^\circ) = \frac{\sqrt{2}}{2} \)

  \( \tan(45^\circ) = 1 \)

• For \(60^\circ\):

  \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \)

  \( \cos(60^\circ) = \frac{1}{2} \)

  \( \tan(60^\circ) = \sqrt{3} \)

Definition and Explanation

Angle of Elevation: The angle of elevation is the angle formed by the line of sight of an observer looking upwards from the horizontal plane. It is the angle between the horizontal and the line of sight to an object above the horizontal plane. For example, when you look at the top of a tall building from the ground, the angle between the horizontal line from your eye level to the top of the building is the angle of elevation.

Angle of Depression: The angle of depression is the angle formed by the line of sight of an observer looking downwards from the horizontal plane. It is the angle between the horizontal and the line of sight to an object below the horizontal plane. For example, when you look down at a car from the top of a hill, the angle between the horizontal line from your eye level to the car is the angle of depression.

How to Calculate

To calculate angles of elevation and depression, trigonometric ratios are used, particularly tangent (tan). The tangent of an angle in a right-angled triangle is the ratio of the opposite side to the adjacent side.

\( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

Where:

• \(\theta\) is the angle of elevation or depression
• Opposite is the height difference between the observer and the object
• Adjacent is the horizontal distance between the observer and the object

Definition

Bearings are a way of describing direction and are typically used in navigation and surveying. A bearing is the angle measured clockwise from the north direction to the direction of the line. Bearings are usually given in degrees and are often expressed as three-figure bearings to avoid ambiguity.

Measuring Bearings

Bearings are typically measured using a protractor or a compass. They are expressed as three-digit numbers to ensure clarity. For instance, a bearing of 60 degrees is written as 060°, and a bearing of 350 degrees is written as 350°.

Examples

Calculating Bearings

To calculate the bearing between two points on a map, you can follow these steps:

1. Identify the points: Determine the starting point and the endpoint.
2. Draw a line: Draw a straight line between the two points.
3. Measure the angle: Use a protractor to measure the angle from the north line (going clockwise) to the line connecting the two points.