Trigonometry Study Notes: NCEA Level 2 Internal (Walker Maths 2.4)

Trigonometry Overview

Introduction to Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It is fundamental in many areas of science, engineering, and everyday problem-solving. The study of trigonometry typically begins with right triangles and extends to more general cases, including non-right triangles, vectors, and applications involving circles.

• Key Terms:

  • Angle: A measure of rotation between two intersecting lines, usually measured in degrees or radians.

  • Triangle: A polygon with three sides and three angles.

  • Trigonometric Functions: Functions that relate the angles of a triangle to the ratios of its sides (sine, cosine, tangent).

• Applications: Used in navigation, physics, engineering, and architecture.

Measuring Angles

Angles are measured in degrees (°) or radians. Understanding how to convert between these units is essential for solving trigonometric problems.

• Degrees: A full circle is 360°.

• Radians: A full circle is $2\pi$ radians.

• Conversion Formula: $\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$

Trigonometric Functions on Right Triangles

In a right triangle, the trigonometric functions relate the angles to the ratios of the sides.

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

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

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

Unit Circle

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

• Coordinates: Any point on the unit circle can be represented as $(\cos(\theta), \sin(\theta))$.

• Key Angles: $0$,$\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, $2\pi$

Graphing Trigonometric Functions

Trigonometric functions can be graphed to show their periodic nature. The graphs of sine, cosine, and tangent functions are commonly used to analyze patterns and solve equations.

• Sine and Cosine: Both have a period of $2\pi$.

• Tangent: Has a period of $\pi$.

Inverse Trigonometric Functions and Basic Trigonometric Equations

Inverse trigonometric functions allow us to find angles when given side ratios. Solving basic trigonometric equations is a key skill in trigonometry.

• Inverse Functions: $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$

• Example Equation: $\sin(\theta) = 0.5 \Rightarrow \theta = \sin^{-1}(0.5)$

Trigonometric Identities and More Equations

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

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

• Angle Sum and Difference: $\sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b)$

Non-Right Triangles

Trigonometry can also be applied to non-right triangles using the Law of Sines and Law of Cosines.

• Law of Sines: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

• Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos(C)$

Vectors

Vectors are quantities that have both magnitude and direction. Trigonometry is used to resolve vectors into components and analyze their properties.

• Vector Components: $\vec{v} = v_x \hat{i} + v_y \hat{j}$

• Magnitude: $|\vec{v}| = \sqrt{v_x^2 + v_y^2}$

Polar Equations

Polar equations describe curves using polar coordinates $(r, \theta)$ instead of Cartesian coordinates $(x, y)$.

• Conversion: $x = r\cos(\theta)$, $y = r\sin(\theta)$

Parametric Equations

Parametric equations express the coordinates of points as functions of a parameter, often used to describe motion.

• Example: $x = f(t)$, $y = g(t)$

Graphing Complex Numbers

Complex numbers can be represented graphically on the complex plane, with the real part on the x-axis and the imaginary part on the y-axis.

• Complex Number: $z = a + bi$

• Magnitude: $|z| = \sqrt{a^2 + b^2}$