Trigonometry Fundamentals

Trigonometry is the branch of mathematics dealing with the relationships between the angles and sides of triangles, especially right-angled triangles. It has important applications in geometry, physics, engineering, and many other fields.

Basic Trigonometric Concepts

Right Triangle and Trigonometric Ratios

In a right-angled triangle, the sides are named relative to an angle \theta as follows:

  • Opposite: the side opposite to angle \theta
  • Adjacent: the side adjacent (next) to angle \theta
  • Hypotenuse: the longest side opposite to the right angle

The basic trigonometric ratios are defined as:

  • \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}
  • \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}
  • \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}

Example 1: Given a right triangle where the side opposite to angle \theta is 3 units, the adjacent side is 4 units, find \sin \theta, \cos \theta, and \tan \theta.

Unit Circle

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. It is a fundamental tool in trigonometry.

For an angle \theta in standard position, the coordinates of the point on the unit circle are:

(x, y) = (\cos \theta, \sin \theta)

This helps extend the definitions of sine and cosine beyond acute angles to all real numbers.

Example 2: Find \sin 60^\circ and \cos 60^\circ using the unit circle.

Trigonometric Identities

These are key formulas involving trig functions that hold true for all values of \theta.

  • \sin^2 \theta + \cos^2 \theta = 1
  • 1 + \tan^2 \theta = \sec^2 \theta
  • 1 + \cot^2 \theta = \csc^2 \theta
  • \sin (A \pm B) = \sin A \cos B \pm \cos A \sin B
  • \cos (A \pm B) = \cos A \cos B \mp \sin A \sin B

Example 3: Verify the identity \sin^2 \theta + \cos^2 \theta = 1 for \theta = 45^\circ.

Solving Triangles

Solving triangles means finding missing sides or angles. The primary methods are:

  • Right triangle trigonometry (using trig ratios)
  • Law of Sines
  • Law of Cosines

Law of Sines

The Law of Sines states:

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Example 4: In triangle ABC, given A=30^\circ, B=45^\circ, and side a=10, find side b.

Law of Cosines

The Law of Cosines states:

c^2 = a^2 + b^2 - 2ab \cos C

Example 5: In triangle ABC, if sides a=7, b=9, and angle C=60^\circ, find side c.

Graphs of Trigonometric Functions

The sine, cosine, and tangent functions are periodic and have characteristic wave shapes.

  • Sine and Cosine: Period = \(2\pi\), range \([-1,1]\)
  • Tangent: Period = \(\pi\), vertical asymptotes where cosine is zero.

These graphs help understand oscillations, waves, and circular motion.

Practice Problems

Problem 1: Calculate \(\sin 45^\circ\), \(\cos 45^\circ\), and \(\tan 45^\circ\).

Problem 2: Given a right triangle with hypotenuse 10 and adjacent side 6, find the angle adjacent to the side.

Problem 3: In triangle ABC, \(A = 40^\circ\), \(B = 60^\circ\), side \(a = 7\). Find side \(b\) using the Law of Sines.