Angles and Sides

In trigonometry, the study of angles and sides forms the foundation of understanding how triangles function. An angle is formed by two rays meeting at a common endpoint called the vertex. The measure of an angle can be in degrees or radians, which are two units of measurement commonly used in trigonometry. The sides of a triangle are classified into three categories: the hypotenuse, the opposite side, and the adjacent side, depending on their position relative to a given angle. For right-angled triangles, the hypotenuse is always the longest side, opposite the right angle.
Example: Consider a right-angled triangle with one angle measuring 30°. If the hypotenuse is 10 units long, then the side opposite the 30° angle can be found using trigonometric ratios such as sine.
Tip: Remember ‘SOHCAHTOA’ to recall the basic trigonometric ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Degree Measure of Angles

The degree measure of an angle is one of the most common ways to quantify angles. A full circle is divided into 360 equal parts, each part representing one degree. This system of measurement is extremely versatile and is used in various fields including navigation, construction, and astronomy.

Example: A right angle measures 90°, and an acute angle is any angle less than 90°, while an obtuse angle is more than 90° but less than 180°.
Tip: To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.
Triangles and Their Angles and Ratios
Triangles are polygons with three sides and three angles. The sum of the angles in any triangle is always 180°. In trigonometry, we often use the angles and side lengths of triangles to calculate unknown values using trigonometric ratios like sine, cosine, and tangent.

Example: In a right-angled triangle, if you know one angle (other than the right angle) and one side, you can find the other sides using trigonometric ratios.
Tip: Use the Law of Sines and the Law of Cosines for non-right triangles to find unknown sides or angles.

Half Chord

The concept of a half chord is closely related to the study of circles in trigonometry. A chord is a line segment with both endpoints on the circle, and a half chord is simply half of this segment. In the context of trigonometry, the half chord can help in determining the sine of an angle, as it represents the vertical distance from the center of the circle to the chord.

Example: For a unit circle (radius = 1), the sine of an angle is equal to the length of the half chord corresponding to that angle.

Tip: Visualize the unit circle to easily remember that the sine of an angle is the y-coordinate of the corresponding point on the circle.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions that describe the relationships between the angles and sides of a triangle. In a right-angled triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse, while the cosine is the ratio of the length of the adjacent side to the hypotenuse.

Example: For an angle of 30° in a right-angled triangle, sine(30°) = 1/2 and cosine(30°) = √3/2.

Tip: Use the mnemonic ‘SOHCAHTOA’ to quickly recall the definitions of sine, cosine, and tangent.

Triangle and Circle

The relationship between triangles and circles is a key concept in trigonometry. A common example is the circumcircle of a triangle, which is a circle that passes through all three vertices of the triangle. The radius of this circumcircle can be found using the formula R = abc/4Δ, where a, b, and c are the sides of the triangle, and Δ is the area of the triangle.

Example: For an equilateral triangle with side length a, the radius of the circumcircle is given by R = a/√3.

Tip: Remember that the angle subtended by a diameter at the circumference of a circle is always 90°, which can help in solving problems related to circles and triangles.

New Measure of Angles

In trigonometry, angles can also be measured in radians, which is another unit of angular measure. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. Radians are particularly useful in calculus and higher-level mathematics because they simplify the relationship between the arc length and the radius.
Example: A full circle has 2π radians, and a right angle is π/2 radians.
Tip: To convert degrees to radians, remember that π radians = 180°. Thus, multiply degrees by π/180 to convert to radians.

Measure of Slant

The measure of slant refers to the inclination or slope of a line or surface. In trigonometry, this is often represented by the tangent of the angle that the line makes with the horizontal. The measure of slant is important in various applications such as determining the steepness of a hill, the incline of a road, or the pitch of a roof.
Example: If a road rises 3 meters for every 4 meters of horizontal distance, the angle of the slope can be found using tan(θ) = 3/4.
Tip: Visualize the slope as a right triangle, where the vertical rise and horizontal run form the opposite and adjacent sides respectively, making it easier to apply trigonometric functions.

Distances and Heights

Calculating distances and heights is one of the most practical applications of trigonometry. By using the angle of elevation or depression along with a known distance or height, one can easily determine the unknown distance or height using trigonometric ratios.
Example: To find the height of a tree when you know the distance from the tree and the angle of elevation, you can use the formula height = distance * tan(θ).
Tip: Remember to always check whether you are using the angle of elevation or depression, as this determines whether you are working above or below the horizontal line.

Angle of Elevation

The angle of elevation is the angle formed by the line of sight of an observer looking at an object above the horizontal level. This angle is crucial in various real-world applications such as architecture, astronomy, and navigation.
Example: If you are standing 50 meters away from a building and the angle of elevation to the top of the building is 30°, you can calculate the height of the building using trigonometric functions.
Tip: To easily remember, think of the angle of elevation as ‘elevating’ your line of sight to look up at something higher than you.

Angle of Depression

The angle of depression is the angle formed between the horizontal line and the line of sight when an observer looks at an object below the horizontal level. This concept is widely used in fields like engineering, aviation, and maritime navigation.
Example: If you are on top of a lighthouse and observe a boat at sea with an angle of depression of 20°, you can use trigonometry to calculate how far the boat is from the base of the lighthouse.
using the tangent of the angle of depression and the known height of the lighthouse.
Tip: Visualize the angle of depression as ‘depressing’ your line of sight to look down at something below your horizontal view.

.