Trigonometry

Definition of an Angle

An angle is formed by two rays (arms) that share a common endpoint (vertex). It represents the amount of rotation between the two rays.

Angles in trigonometry are typically measured:

• From a reference line (usually the positive x-axis)
• In a specified rotation direction (clockwise or counterclockwise)

Definition of Trigonometric Functions

Trigonometric functions relate the angles of a right-angled triangle to the ratios of its sides. These functions are fundamental in trigonometry, geometry, and physics.

Trigonometric Ratios in a Right-Angled Triangle

For a right-angled triangle with angle \( \theta \), we define:

• Sine (\( \sin \theta \)) = Opposite side / Hypotenuse
• Cosine (\( \cos \theta \)) = Adjacent side / Hypotenuse
• Tangent (\( \tan \theta \)) = Opposite side / Adjacent side
• Cosecant (\( \csc \theta \)) = Hypotenuse / Opposite side
• Secant (\( \sec \theta \)) = Hypotenuse / Adjacent side
• Cotangent (\( \cot \theta \)) = Adjacent side / Opposite side

Summary Table

Quadrants of the Cartesian Plane

The Cartesian plane is divided into four quadrants, each characterized by the signs of the x and y coordinates:

• Quadrant I: x > 0, y > 0 (Top Right)
• Quadrant II: x < 0, y > 0 (Top Left)
• Quadrant III: x < 0, y < 0 (Bottom Left)
• Quadrant IV: x > 0, y < 0 (Bottom Right)

CAST Rule for Signs of Trigonometric Functions

The CAST rule helps remember which trigonometric functions are positive in each quadrant:

• Quadrant I (All): All trigonometric functions (sin, cos, tan, csc, sec, cot) are positive.
• Quadrant II (Sine): Sine (sin) and its reciprocal, cosecant (csc), are positive.
• Quadrant III (Tangent): Tangent (tan) and its reciprocal, cotangent (cot), are positive.
• Quadrant IV (Cosine): Cosine (cos) and its reciprocal, secant (sec), are positive.

This can be summarized by the acronym CAST, starting from Quadrant IV and moving counterclockwise.

Understanding Reduction Formulas

Reduction formulas are used to express trigonometric functions of angles greater than 90° in terms of their acute angle (less than 90°) values. This simplifies calculations and analysis by relating angles in different quadrants to reference angles in the first quadrant.

General Approach

To apply reduction formulas:

• Identify the quadrant in which the angle lies.
• Express the angle as a multiple of 90° (π/2 radians) plus or minus an acute angle (θ).
• Key Tip:
  • If the angle is of the form \(90^\circ \pm \theta\) or \(270^\circ \pm \theta\), the trigonometric function changes to its complementary function (sin ↔ cos, tan ↔ cot, sec ↔ csc).
  • If the angle is of the form \(0^\circ \pm \theta\), \(180^\circ \pm \theta\), or \(360^\circ \pm \theta\), the trigonometric function remains the same.
  The sign of the resulting trigonometric function depends on the original function's sign in that quadrant (use the CAST rule).
• Apply the appropriate reduction formula based on the quadrant and trigonometric function.

Formulas Overview

The following tables summarize reduction formulas for angles in the form of \( n \cdot 90^{\circ} \pm \theta \), where \( n \) is an integer and \( \theta \) is an acute angle.

Quadrant II (90° < angle < 180°)

Quadrant III (180° < angle < 270°)

Quadrant IV (270° < angle < 360°)

Examples

• sin(120°) = sin(180° - 60°) = sin(60°) = √3/2
• cos(225°) = cos(180° + 45°) = -cos(45°) = -1/√2
• tan(300°) = tan(360° - 60°) = -tan(60°) = -√3

Overview

These formulas allow you to convert between sums and products of trigonometric functions, which can be useful in various applications, including simplifying expressions and solving equations.

Sum-to-Product Formulas

• \[ \sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \]
• \[ \sin A - \sin B = 2 \cos \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right) \]
• \[ \cos A + \cos B = 2 \cos \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \]
• \[ \cos A - \cos B = -2 \sin \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right) \]

Product-to-Sum Formulas

• \[ \sin A \cos B = \frac{1}{2} \left[ \sin(A + B) + \sin(A - B) \right] \]
• \[ \cos A \sin B = \frac{1}{2} \left[ \sin(A + B) - \sin(A - B) \right] \]
• \[ \cos A \cos B = \frac{1}{2} \left[ \cos(A + B) + \cos(A - B) \right] \]
• \[ \sin A \sin B = \frac{1}{2} \left[ \cos(A - B) - \cos(A + B) \right] \]

Derivation from Addition Formulas

Double angle formulas express trigonometric functions of \(2\theta\) in terms of trigonometric functions of \(\theta\). These can be derived from the addition formulas by setting \(A = B = \theta\).

Sine Double Angle Formula

Using \(\sin(A + B) = \sin A \cos B + \cos A \sin B\), let \(A = B = \theta\):

\[ \sin(2\theta) = \sin(\theta + \theta) = \sin \theta \cos \theta + \cos \theta \sin \theta = 2 \sin \theta \cos \theta \]

Cosine Double Angle Formulas

Using \(\cos(A + B) = \cos A \cos B - \sin A \sin B\), let \(A = B = \theta\):

\[ \cos(2\theta) = \cos(\theta + \theta) = \cos \theta \cos \theta - \sin \theta \sin \theta = \cos^2 \theta - \sin^2 \theta \]

Alternative forms using \(\sin^2 \theta + \cos^2 \theta = 1\):

• \[ \cos(2\theta) = 2 \cos^2 \theta - 1 \]
• \[ \cos(2\theta) = 1 - 2 \sin^2 \theta \]

Tangent Double Angle Formula

Using \(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\), let \(A = B = \theta\):

\[ \tan(2\theta) = \tan(\theta + \theta) = \frac{\tan \theta + \tan \theta}{1 - \tan \theta \tan \theta} = \frac{2 \tan \theta}{1 - \tan^2 \theta} \]

Overview

Half angle formulas allow you to express trigonometric functions of \(\frac{\theta}{2}\) in terms of trigonometric functions of \(\theta\). They are derived from the double angle formulas and are useful in simplifying expressions and solving equations.

Sine Half Angle Formula

From \(\cos(2\theta) = 1 - 2\sin^2(\theta)\), let \(\theta = \frac{\alpha}{2}\):

\[ \cos(\alpha) = 1 - 2\sin^2\left(\frac{\alpha}{2}\right) \]

Solving for \(\sin\left(\frac{\alpha}{2}\right)\):

\[ \sin\left(\frac{\alpha}{2}\right) = \pm \sqrt{\frac{1 - \cos(\alpha)}{2}} \]

Cosine Half Angle Formula

From \(\cos(2\theta) = 2\cos^2(\theta) - 1\), let \(\theta = \frac{\alpha}{2}\):

\[ \cos(\alpha) = 2\cos^2\left(\frac{\alpha}{2}\right) - 1 \]

Solving for \(\cos\left(\frac{\alpha}{2}\right)\):

\[ \cos\left(\frac{\alpha}{2}\right) = \pm \sqrt{\frac{1 + \cos(\alpha)}{2}} \]

Tangent Half Angle Formulas

Using \(\tan\left(\frac{\alpha}{2}\right) = \frac{\sin\left(\frac{\alpha}{2}\right)}{\cos\left(\frac{\alpha}{2}\right)}\) and the above formulas:

\[ \tan\left(\frac{\alpha}{2}\right) = \pm \sqrt{\frac{1 - \cos(\alpha)}{1 + \cos(\alpha)}} \]

Alternative forms:

• \[ \tan\left(\frac{\alpha}{2}\right) = \frac{\sin(\alpha)}{1 + \cos(\alpha)} \]
• \[ \tan\left(\frac{\alpha}{2}\right) = \frac{1 - \cos(\alpha)}{\sin(\alpha)} \]

Note on Signs

The sign (\(\pm\)) in front of the square root depends on the quadrant in which \(\frac{\alpha}{2}\) lies. You need to determine whether the trigonometric function is positive or negative in that quadrant.

Overview

In many practical applications, approximate values of trigonometric functions are used, especially for angles that are not standard angles (0°, 30°, 45°, 60°, 90°). These approximations are often based on series expansions or numerical methods.

Small Angle Approximations

For small angles (θ close to 0 radians or 0°), the following approximations are commonly used:

• \[ \sin \theta \approx \theta \] (where θ is in radians)
• \[ \cos \theta \approx 1 - \frac{\theta^2}{2} \] (where θ is in radians)
• \[ \tan \theta \approx \theta \] (where θ is in radians)

These approximations are derived from the Maclaurin series expansions of the trigonometric functions.

Linear Interpolation

Linear interpolation can be used to estimate values between known values. For example, if you know sin(30°) and sin(45°), you can estimate sin(35°) by linear interpolation:

\[ \sin(35^\circ) \approx \sin(30^\circ) + \frac{35^\circ - 30^\circ}{45^\circ - 30^\circ} (\sin(45^\circ) - \sin(30^\circ)) \]

\[ \sin(35^\circ) \approx 0.5 + \frac{5}{15} (\frac{\sqrt{2}}{2} - 0.5) \approx 0.5707 \]

Using Calculators and Software

In practice, calculators and software are commonly used to find approximate values of trigonometric functions. These tools use numerical methods to compute the values to a high degree of accuracy.

Example

Find an approximate value of sin(5°) using the small angle approximation:

Convert 5° to radians: \[ \theta = 5^\circ \times \frac{\pi}{180} \approx 0.0873 \text{ radians} \]

Therefore, \[ \sin(5^\circ) \approx 0.0873 \]

The actual value (from a calculator) is approximately 0.08716, so the approximation is quite accurate for small angles.

Overview

These relations connect the sides and angles of a triangle, allowing you to solve for unknown elements given sufficient information.

Angle Sum Property

In any triangle, the sum of the angles is 180° (π radians):

\[ A + B + C = 180^\circ \]

Sine Rule (Law of Sines)

Relates the sides of a triangle to the sines of their opposite angles:

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

where a, b, c are the lengths of the sides opposite to angles A, B, C, respectively.

Cosine Rule (Law of Cosines)

Relates the sides and angles in a triangle and is useful for finding an angle if all three sides are known or finding a side if two sides and the included angle are known:

• \[ a^2 = b^2 + c^2 - 2bc \cos A \]
• \[ b^2 = a^2 + c^2 - 2ac \cos B \]
• \[ c^2 = a^2 + b^2 - 2ab \cos C \]

Projection Formulas

These formulas express each side of a triangle in terms of the other two sides and the cosines of the adjacent angles.

• \( a = b \cos C + c \cos B \)
• \( b = c \cos A + a \cos C \)
• \( c = a \cos B + b \cos A \)

Example

In triangle ABC, if A = 60°, B = 45°, and a = 10, find b using the Sine Rule:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} \Rightarrow \frac{10}{\sin 60^\circ} = \frac{b}{\sin 45^\circ} \]

\[ b = \frac{10 \times \sin 45^\circ}{\sin 60^\circ} = \frac{10 \times \frac{\sqrt{2}}{2}}{\frac{\sqrt{3}}{2}} = 10 \times \frac{\sqrt{2}}{\sqrt{3}} \approx 8.16 \]