Introduction

Trigonometry is a branch of mathematics that deals with the study of angles and the relationships between the lengths and angles of triangles. It is an essential topic in mathematics as it has applications in various fields such as physics, engineering, and architecture. In this guide, we will explore the key concepts of trigonometry that Grade 11 CBC learners need to understand.

Basic Trigonometric Ratios

Trigonometry involves three primary trigonometric ratios: sine ($\sin$), cosine ($\cos$), and tangent ($\tan$). These ratios are defined based on the sides of a right-angled triangle.

1. Sine ($\sin$): In a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.

   • $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$

   Example: Given a right triangle with an angle of 30 degrees, and the opposite side measuring 4 cm and the hypotenuse measuring 8 cm, find the sine of the angle.

   $$\sin(30^\circ) = \frac{4}{8} = 0.5$$

2. Cosine ($\cos$): The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse.

   • $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

   Example: In a triangle with an angle of 45 degrees, if the adjacent side is 3 cm and the hypotenuse is 5 cm, calculate the cosine of the angle.

   $$\cos(45^\circ) = \frac{3}{5} = 0.6$$

3. Tangent ($\tan$): The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the adjacent side.

   • $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

   Example: Find the tangent of an angle of 60 degrees in a triangle where the opposite side is 5 cm and the adjacent side is 5√3 cm.

   $$\tan(60^\circ) = \frac{5}{5\sqrt{3}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$$

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. Some common identities include:

1. Pythagorean Identity: In a right triangle, the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the hypotenuse.

   • $\sin^2(\theta) + \cos^2(\theta) = 1$

2. Reciprocal Identities:

   • $\csc(\theta) = \frac{1}{\sin(\theta)}$
   • $\sec(\theta) = \frac{1}{\cos(\theta)}$
   • $\cot(\theta) = \frac{1}{\tan(\theta)}$

3. Even-Odd Identities:

   • $\sin(-\theta) = -\sin(\theta)$
   • $\cos(-\theta) = \cos(\theta)$
   • $\tan(-\theta) = -\tan(\theta)$

Trigonometric Functions of Special Angles

Special angles such as 30°, 45°, and 60° have exact values for their trigonometric functions. These values are often used in calculations.

1. For a 30-60-90 triangle:

   • $\sin(30^\circ) = \frac{1}{2}$, $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, $\tan(30^\circ) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$
   • $\sin(60^\circ) = \frac{\sqrt{3}}{2}$, $\cos(60^\circ) = \frac{1}{2}$, $\tan(60^\circ) = \sqrt{3}$

2. For a 45-45-90 triangle:

   • $\sin(45^\circ) = \cos(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
   • $\tan(45^\circ) = 1$

Unit Circle and Trigonometric Functions

The unit circle is a circle with a radius of 1 unit centered at the origin of a coordinate plane. It is used to define trigonometric functions for all real numbers.

1. Definition of Trigonometric Functions on the Unit Circle:

   • Given an angle $\theta$ in standard position on the unit circle, the coordinates of the point where the terminal side intersects the circle are $(\cos(\theta), \sin(\theta))$.

2. Trigonometric Functions of Quadrantal Angles:

   • In the first quadrant, all trigonometric ratios are positive.
   • In the second quadrant, only sine is positive.
   • In the third quadrant, only tangent is positive.
   • In the fourth quadrant, only cosine is positive.

Common Mistakes

• Forgetting to convert angles to the correct unit (degrees or radians) before applying trigonometric functions.
• Misidentifying the opposite, adjacent, and hypotenuse sides in a triangle.
• Incorrectly applying trigonometric identities and ratios in complex problems.
• Failing to simplify expressions involving trigonometric functions.

Key Points

• Trigonometry deals with the study of angles and the relationships between the sides of a triangle.
• The main trigonometric ratios are sine, cosine, and tangent.
• Trigonometric identities, special angles, and the unit circle are essential concepts in trigonometry.

Practice Questions

1. Calculate the value of $\sin(45^\circ) + \cos(45^\circ)$.

   Solution: $\sin(45^\circ) + \cos(45^\circ) = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \sqrt{2}$

2. If $\sin(\theta) = \frac{3}{5}$ and $\theta$ is in the second quadrant, find $\cos(\theta)$.

   Solution: Since $\theta$ is in the second quadrant, cosine is negative. Using Pythagorean identity, $\cos(\theta) = -\sqrt{1 - \sin^2(\theta)} = -\frac{4}{5}$

3. Given $\tan(\alpha) = \frac{5}{12}$, find the value of $\csc(\alpha)$.

   Solution: $\csc(\alpha) = \frac{1}{\sin(\alpha)} = \frac{1}{\frac{5}{13}} = \frac{13}{5}$

4. Determine the value of $\sin\left(\frac{5\pi}{6}\right)$.

   Solution: $\sin\left(\frac{5\pi}{6}\right) = -\frac{1}{2}$

5. If $\cos(\beta) = -\frac{3}{5}$ and $\beta$ is in the third quadrant, find $\tan(\beta)$.

   Solution: Since $\beta$ is in the third quadrant, tangent is positive. Using Pythagorean identity, $\tan(\beta) = \frac{\sin(\beta)}{\cos(\beta)} = \frac{4}{3}$