Angles and the Trigonometric Functions: Comprehensive Study Notes

Angles and the Trigonometric Functions

Introduction to Angles

Angles are fundamental in trigonometry, representing the amount of rotation between two rays sharing a common endpoint (the vertex). Understanding angles and their properties is essential for studying trigonometric functions and their applications.

• Angle: The figure formed by two rays (the initial and terminal sides) sharing a common endpoint (the vertex).

• Standard Position: An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis.

• Positive Angles: Generated by counterclockwise rotation from the initial side.

• Negative Angles: Generated by clockwise rotation from the initial side.

Types of Angles:

• Acute: $0^ ext{o} < \theta < 90^ ext{o}$

• Right: $\theta = 90^ ext{o}$

• Obtuse: $90^ ext{o} < \theta < 180^ ext{o}$

• Straight: $\theta = 180^ ext{o}$

Example: Sketching angles such as $60^ ext{o}$, $150^ ext{o}$, and $-90^ ext{o}$ in standard position.

Coterminal Angles

Coterminal angles share the same initial and terminal sides but may have different measures. They are found by adding or subtracting multiples of $360^ ext{o}$ (or $2\pi$ radians).

• Coterminal Angle Formula: $\theta + 360^ ext{o}k$, where $k$ is any integer.

• To find a positive coterminal angle, add $360^ ext{o}$ until the result is positive and less than $360^ ext{o}$.

Example: Find a coterminal angle for $390^ ext{o}$, $-270^ ext{o}$, and $1000^ ext{o}$.

Special Right Triangles

Special right triangles, such as the $45^ ext{o}$-$45^ ext{o}$-$90^ ext{o}$ and $30^ ext{o}$-$60^ ext{o}$-$90^ ext{o}$ triangles, have side lengths that follow specific ratios, making them useful for solving trigonometric problems without a calculator.

45-45-90 Triangles

• Both legs are congruent.

• Hypotenuse: $\text{leg} \times \sqrt{2}$

• Side ratios: $1:1:\sqrt{2}$

30-60-90 Triangles

• Short leg (opposite $30^ ext{o}$): $x$

• Long leg (opposite $60^ ext{o}$): $x\sqrt{3}$

• Hypotenuse: $2x$

• Side ratios: $1:\sqrt{3}:2$

Example: Solve for unknown sides in special right triangles using these ratios.

Trigonometric Functions in Right Triangles

Trigonometric functions relate the angles of a right triangle to the ratios of its sides. The three primary functions are sine, cosine, and tangent.

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

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

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

Reciprocal functions:

• $\csc \theta = \frac{1}{\sin \theta}$

• $\sec \theta = \frac{1}{\cos \theta}$

• $\cot \theta = \frac{1}{\tan \theta}$

Mnemonic: SOH-CAH-TOA helps remember the definitions.

Degrees and Radians

Angles can be measured in degrees or radians. Radians are based on the radius of a circle and are the standard unit in higher mathematics.

• $180^ ext{o} = \pi$ radians

• To convert degrees to radians: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$

• To convert radians to degrees: $\text{degrees} = \text{radians} \times \frac{180}{\pi}$

Example: Convert $120^ ext{o}$ to radians and $\frac{5\pi}{6}$ to degrees.

Complementary and Supplementary Angles

Complementary angles sum to $90^ ext{o}$, while supplementary angles sum to $180^ ext{o}$.

• Complementary: $A + B = 90^ ext{o}$

• Supplementary: $A + B = 180^ ext{o}$

Example: Find the complement and supplement of a given angle.

Cofunctions of Complementary Angles

Cofunction identities relate trigonometric functions of complementary angles:

• $\sin A = \cos (90^ ext{o} - A)$

• $\tan A = \cot (90^ ext{o} - A)$

• $\sec A = \csc (90^ ext{o} - A)$

Example: Express $\sin(37^ ext{o})$ in terms of its cofunction.

Solving Right Triangles

To solve right triangles, use trigonometric ratios and the Pythagorean Theorem ($a^2 + b^2 = c^2$) to find missing sides or angles.

• Identify known sides and angles.

• Choose the appropriate trigonometric function.

• Solve for the unknown value.

Example: Given a right triangle with sides 3 and 4, find the missing angle.

Common Values of Sine, Cosine, and Tangent

Memorize the sine, cosine, and tangent values for $30^ ext{o}$, $45^ ext{o}$, and $60^ ext{o}$ for quick reference:

• $\sin 30^ ext{o} = \frac{1}{2}$, $\cos 30^ ext{o} = \frac{\sqrt{3}}{2}$, $\tan 30^ ext{o} = \frac{1}{\sqrt{3}}$

• $\sin 45^ ext{o} = \frac{\sqrt{2}}{2}$, $\cos 45^ ext{o} = \frac{\sqrt{2}}{2}$, $\tan 45^ ext{o} = 1$

• $\sin 60^ ext{o} = \frac{\sqrt{3}}{2}$, $\cos 60^ ext{o} = \frac{1}{2}$, $\tan 60^ ext{o} = \sqrt{3}$

The Unit Circle

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

• Equation: $x^2 + y^2 = 1$

• Coordinates $(x, y)$ on the unit circle correspond to $(\cos \theta, \sin \theta)$ for an angle $\theta$.

Example: Find $\sin 120^ ext{o}$, $\cos 225^ ext{o}$, and $\tan 330^ ext{o}$ using the unit circle.

Reference Angles

A reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. It helps in finding trigonometric values for any angle.

• Reference angle for $\theta$ in quadrant II: $180^ ext{o} - \theta$

• Reference angle for $\theta$ in quadrant III: $\theta - 180^ ext{o}$

• Reference angle for $\theta$ in quadrant IV: $360^ ext{o} - \theta$

Example: Find the reference angle for $210^ ext{o}$.

Trig Values in Quadrants II, III, IV

Trig values change sign depending on the quadrant:

• Quadrant I: All positive

• Quadrant II: Sine positive

• Quadrant III: Tangent positive

• Quadrant IV: Cosine positive

Mnemonic: "All Students Take Calculus" (ASTC)

Reciprocal Trig Functions on the Unit Circle

Secant, cosecant, and cotangent are defined as reciprocals of cosine, sine, and tangent, respectively, and can be evaluated using the unit circle.

• $\sec \theta = \frac{1}{\cos \theta}$

• $\csc \theta = \frac{1}{\sin \theta}$

• $\cot \theta = \frac{1}{\tan \theta}$

Tables: Trig Function Values for Special Triangles

The following tables summarize the trigonometric function values for $45^ ext{o}$-$45^ ext{o}$-$90^ ext{o}$ and $30^ ext{o}$-$60^ ext{o}$-$90^ ext{o}$ triangles:

Function	45°-45°-90°	30°-60°-90°
sin	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$ (30°), $\frac{\sqrt{3}}{2}$ (60°)
cos	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$ (30°), $\frac{1}{2}$ (60°)
tan	$1$	$\frac{1}{\sqrt{3}}$ (30°), $\sqrt{3}$ (60°)
csc	$\sqrt{2}$	$2$ (30°), $\frac{2}{\sqrt{3}}$ (60°)
sec	$\sqrt{2}$	$\frac{2}{\sqrt{3}}$ (30°), $2$ (60°)
cot	$1$	$\sqrt{3}$ (30°), $\frac{1}{\sqrt{3}}$ (60°)

Solving Trigonometric Equations Using Cofunction Identities

To solve equations involving cofunction identities, rewrite one side using the identity and set the arguments equal.

• Example: $\sin(x) = \cos(90^ ext{o} - x)$

Using Calculators for Trig Functions

When evaluating trigonometric functions, ensure your calculator is in the correct mode (degrees or radians). For inverse functions, use the $\sin^{-1}$, $\cos^{-1}$, or $\tan^{-1}$ keys to find angles.

Applications of Right Triangle Trigonometry

Trigonometry is used to solve real-world problems involving heights, distances, and angles of elevation or depression.

• Example: Calculating the height of a lighthouse or the angle of a hiking path using trigonometric ratios.

Additional info: These notes are based on the content of a Pearson+ Study Prep guide for Trigonometry, Chapter 1, and are structured to provide a comprehensive overview suitable for college-level exam preparation.