Trigonometry: Class Notes

Trigonometry is the branch of mathematics that deals with the relationships between the angles
and sides of triangles. It is essential for studying properties of triangles, especially right-angled
triangles, and has wide applications in science, engineering, and architecture.

Basic Trigonometric Ratios

In a right-angled triangle, where one of the angles is 90°, the three sides are:

1. Hypotenuse (h): The longest side, opposite the right angle.

2. Opposite Side (o): The side opposite the angle of interest.

3. Adjacent Side (a): The side next to the angle of interest (not the hypotenuse).

The primary trigonometric ratios are:

 Sine (sin) of an angle θ\theta =

OppositeHypotenuse\frac{\text{Opposite}}{\text{Hypotenuse}}

 Cosine (cos) of an angle θ\theta =

AdjacentHypotenuse\frac{\text{Adjacent}}{\text{Hypotenuse}}

 Tangent (tan) of an angle θ\theta = OppositeAdjacent\frac{\text{Opposite}}{\text{Adjacent}}

These ratios are used to find unknown sides or angles in right-angled triangles.

Trigonometric Functions

Trigonometric functions can be used for any angle, not just angles in right-angled triangles. The
basic functions (sine, cosine, tangent) are extended to all angles, often using the unit circle. Here's
a quick overview:

1. Sine (sin): For any angle θ\theta, sine is the y-coordinate of a point on the unit circle
corresponding to θ\theta.

2. Cosine (cos): Cosine is the x-coordinate of the point corresponding to the angle θ\theta on
the unit circle.

3. Tangent (tan): The tangent of an angle θ\theta is the ratio of sine to cosine:
tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}.

The Reciprocal Trigonometric Functions

These functions are the reciprocals of sine, cosine, and tangent:

1. Cosecant (csc): The reciprocal of sine. csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}

 2. Secant (sec): The reciprocal of cosine. sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}

3. Cotangent (cot): The reciprocal of tangent. cot⁡θ=1tan⁡θ=cos⁡θsin⁡θ\cot \theta =

\frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}

Trigonometric Identities

These are fundamental relationships between the trigonometric functions:

1. Pythagorean Identity: sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

2. Reciprocal Identities: csc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=1tan⁡θ\csc \theta =

\frac{1}{\sin \theta}, \sec \theta = \frac{1}{\cos \theta}, \cot \theta = \frac{1}{\tan \theta}

3. Quotient Identities: tan⁡θ=sin⁡θcos⁡θ,cot⁡θ=cos⁡θsin⁡θ\tan \theta = \frac{\sin

\theta}{\cos \theta}, \cot \theta = \frac{\cos \theta}{\sin \theta}

4. Sum and Di erence Formulas:

o For sine: sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A + B) = \sin A \cos B + \cos A

\sin B sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin(A - B) = \sin A \cos B - \cos A \sin B

o For cosine: cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A + B) = \cos A \cos B - \sin

A \sin B cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos(A - B) = \cos A \cos B + \sin A
\sin B

o For tangent: tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B\tan(A + B) = \frac{\tan A +

\tan B}{1 - \tan A \tan B} tan⁡(A−B)=tan⁡A−tan⁡B1+tan⁡Atan⁡B\tan(A - B) =
\frac{\tan A - \tan B}{1 + \tan A \tan B}

Unit Circle and Trigonometric Functions

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. Each point
on the unit circle corresponds to an angle θ\theta and has coordinates (x,y)(x, y) where:

 x=cos⁡θx = \cos \theta

 y=sin⁡θy = \sin \theta

 Angle in Degrees: 360° corresponds to a full circle.

 Angle in Radians: 2π radians correspond to a full circle.

The unit circle helps in understanding trigonometric functions for all angles (not just those in right-
angled triangles).

Trigonometric Ratios for Common Angles

Here are the values for the sine, cosine, and tangent functions at key angles:

Angle (°) 0° 30° 45° 60° 90° 120° 135° 150° 180° 270° 360°

sin(θ) 0 1/2 √2/2 √3/2 1 √3/2 √2/2 1/2 0 -1 0

cos(θ) 1 √3/2 √2/2 1/2 0 -1/2 -√2/2 -√3/2 -1 0 1

tan(θ) 0 1/√3 1 √3 ∞ -√3 -1 -1/√3 0 0 0

Applications of Trigonometry

1. Solving Right-Angled Triangles: Trigonometry is used to find missing angles or sides in

right-angled triangles using the basic trigonometric ratios.

2. Navigation and Surveying: Trigonometry helps calculate distances and angles between
landmarks when navigating or surveying land.

3. Physics: Trigonometry is used in mechanics (to calculate forces, angles, etc.), waves (in
oscillations and wave motion), and electromagnetism.

4. Engineering: In designing mechanical structures, trigonometric calculations are crucial for

angles, forces, and distances.

5. Astronomy: Trigonometry helps in calculating distances between celestial bodies using

angles and height measurements.

Inverse Trigonometric Functions

To find an angle when the trigonometric ratio is given, we use inverse functions:

 sin⁡−1(x)\sin^{-1}(x) or arcsin(x): Finds the angle whose sine is xx.

 cos⁡−1(x)\cos^{-1}(x) or arccos(x): Finds the angle whose cosine is xx.

 tan⁡−1(x)\tan^{-1}(x) or arctan(x): Finds the angle whose tangent is xx.

Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions. To solve these equations,
you may need to use trigonometric identities, algebraic manipulation, and inverse trigonometric
functions. For example:

1. Basic Example: sin⁡θ=1/2\sin \theta = 1/2

o Solve for θ\theta within a specific range (e.g., 0° to 360° or 0 to 2π2\pi).

2. More Complex Example: 2cos⁡θ−1=02 \cos \theta - 1 = 0

 o Solve for θ\theta using algebraic techniques and trigonometric identities.

Conclusion

Trigonometry is a powerful mathematical tool used in many fields. Understanding the basic
trigonometric ratios, identities, and their applications can help solve a wide variety of problems,
from calculating angles and distances in geometry to understanding wave motion and oscillations
in physics. Mastery of these concepts forms the foundation for more advanced study in
mathematics and science.