Trigonometry (Forms 5–6)

Arts 1–4

Foreign Languages 1–4

Covers trigonometric identities, equations, and applications including bearings and real-world problems.

📘 Topic Summary

Trigonometry is a branch of mathematics that deals with the relationships between the angles and side lengths of triangles. In this study guide, we will explore the forms 5-6 trigonometry topics, including trigonometric identities, equations, and applications such as bearings and real-world problems.

📖 Glossary

Sine: The ratio of the opposite side to the hypotenuse in a right triangle.
Cosine: The ratio of the adjacent side to the hypotenuse in a right triangle.
Tangent: The ratio of the opposite side to the adjacent side in a right triangle.
Pythagorean Identity: sin^2(A) + cos^2(A) = 1
Trigonometric Functions: Functions that relate the angles and side lengths of triangles

⭐ Key Points

The sine, cosine, and tangent functions are used to solve triangular problems.
The Pythagorean identity is used to simplify trigonometric expressions.
Trigonometry has many real-world applications, including navigation and physics.
The unit circle is a useful tool for visualizing trigonometric relationships.
Trigonometric identities can be used to solve equations and prove theorems.

🔍 Subtopics

Introduction to Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It involves the use of trigonometric functions such as sine, cosine, and tangent to solve problems involving right-angled triangles. These functions can be used to calculate distances, angles, and heights in various real-world applications.

Trigonometric Identities

The Pythagorean identity sin^2(A) + cos^2(A) = 1 is a fundamental concept in trigonometry. Other important identities include the sum and difference formulas for sine, cosine, and tangent: sin(A+B) = sin(A)cos(B) + cos(A)sin(B), and tan(A+B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B)). These identities can be used to simplify trigonometric expressions and solve equations.

Solving Triangular Problems

To solve triangular problems, we can use the sine, cosine, or tangent ratio. For example, if we know the length of two sides and the measure of one angle, we can use the sine or cosine ratio to find the third side. Alternatively, we can use the Pythagorean theorem to find the hypotenuse of a right-angled triangle.

Applications of Trigonometry

Trigonometry has many practical applications in fields such as navigation, physics, and engineering. For example, trigonometric functions can be used to calculate distances and directions between two points on the surface of the Earth. In physics, trigonometry is used to describe the motion of objects in terms of position, velocity, and acceleration.

Trigonometric Functions and Equations

The sine, cosine, and tangent functions are periodic with a period of 2π radians. The graphs of these functions have an amplitude that depends on the coefficient of the function. Trigonometric equations can be solved using various methods such as factoring, quadratic formula, or graphical methods.

Graphs of Trigonometric Functions

The graphs of sine, cosine, and tangent functions are sinusoidal curves that repeat every 2π radians. The amplitude and period of these curves can be adjusted by multiplying the function by a coefficient or adding a phase shift. These graphs have many real-world applications in fields such as physics and engineering.

Trigonometry in Real-World Problems

Trigonometry is used to solve problems involving right-angled triangles, including calculating distances, heights, and angles. It is also used in navigation systems such as GPS, where it helps determine the user's location and direction of travel.

Common Mistakes and Misconceptions

One common mistake when working with trigonometric functions is to confuse the sine, cosine, and tangent ratios. Another mistake is to forget to consider the quadrant in which an angle lies. It is also important to remember that trigonometric identities are only true for angles measured in radians.

Practice Problems and Exercises

Solve the following problems: Find the length of the hypotenuse of a right-angled triangle with legs of length 3 cm and 4 cm. Use the sine ratio to find the height of a building that casts a shadow of 20 meters when the angle of elevation is 30°.

Subjects