Precalculus Study Guide: Angles, Trigonometry, and Applications

Angles and Their Measurement

Standard Position and Quadrants

Angles in the coordinate plane are often drawn in standard position, where the vertex is at the origin and the initial side lies along the positive x-axis. The position of the terminal side determines the quadrant in which the angle lies.

• Quadrant I: Angles between 0° and 90°

• Quadrant II: Angles between 90° and 180°

• Quadrant III: Angles between 180° and 270°

• Quadrant IV: Angles between 270° and 360°

Example: An angle of 120° lies in Quadrant II.

Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but may have different measures. To find a coterminal angle, add or subtract multiples of 360° (or $2\pi$ radians).

• Formula: $\theta_{coterminal} = \theta + 360^\circ \times n$ or $\theta + 2\pi n$ (where n is an integer)

• Example: The least positive coterminal angle for $-144^\circ$ is $216^\circ$.

Angle Conversion: Degrees and Radians

Converting Between Degrees and Radians

Angles can be measured in degrees or radians. The conversion between these units is essential in trigonometry.

• Degrees to Radians: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$

• Radians to Degrees: $\text{degrees} = \text{radians} \times \frac{180}{\pi}$

• Example: $60^\circ = 60 \times \frac{\pi}{180} = \frac{\pi}{3}$ radians

Arc Length and Sector Area

Arc Length

The arc length of a circle is the distance along the curved part of the circle, calculated using the radius and the central angle in radians.

• Formula: $s = r\theta$ (where s is arc length, r is radius, $\theta$ is angle in radians)

• Example: For a radius of 5 ft and central angle of 1 radian, $s = 5 \times 1 = 5$ ft.

Sector Area

The area of a sector is a portion of the circle's area, determined by the central angle.

• Formula: $A = \frac{1}{2} r^2 \theta$

• Example: For $r = 3$ cm and $\theta = 2$ radians, $A = \frac{1}{2} \times 9 \times 2 = 9$ cm2

Trigonometric Functions and Right Triangles

Basic Trigonometric Ratios

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: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

• Example: In a triangle with sides 8, 15, and 17, $\sin \theta = \frac{8}{17}$

Pythagorean Theorem

The Pythagorean Theorem is used to find the missing side of a right triangle.

• Formula: $a^2 + b^2 = c^2$ (where c is the hypotenuse)

• Example: If $a = 5$, $b = 12$, then $c = \sqrt{5^2 + 12^2} = 13$

Unit Circle and Trigonometric Values

Unit Circle

The unit circle is a circle of radius 1 centered at the origin. The coordinates of a point on the unit circle correspond to the cosine and sine of the angle.

• Coordinates: $(\cos \theta, \sin \theta)$

• Example: For $\theta = 0$, the point is (1, 0)

Reference Angles

A reference angle is the smallest angle between the terminal side of a given angle and the x-axis.

• How to Find: Subtract the angle from the nearest x-axis (0°, 180°, 360°)

• Example: The reference angle for $210^\circ$ is $30^\circ$

Applications: Distance and Similar Triangles

Law of Cosines and Law of Sines

These laws are used to solve triangles that are not right triangles.

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

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

Similar Triangles

Triangles are similar if their corresponding angles are equal and their sides are proportional.

• Proportionality Constant: If the sides of the larger triangle are k times the sides of the smaller triangle, then $\text{side}_{\text{large}} = k \times \text{side}_{\text{small}}$

• Example: If $k = \sqrt{5}$ and a side of the small triangle is 2, then the corresponding side of the large triangle is $2\sqrt{5}$

Trigonometric Identities

Basic Identities

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

• Pythagorean Identity: $\sin^2 x + \cos^2 x = 1$

• Quotient Identity: $\tan x = \frac{\sin x}{\cos x}$

• Reciprocal Identities: $\csc x = \frac{1}{\sin x}$, $\sec x = \frac{1}{\cos x}$, $\cot x = \frac{1}{\tan x}$

Summary Table: Quadrants and Trigonometric Signs

Quadrant	Angle Range	sin	cos	tan
I	0° to 90°	+	+	+
II	90° to 180°	+	-	-
III	180° to 270°	-	-	+
IV	270° to 360°	-	+	-

Additional info:

• Some questions involve practical applications such as finding distances using trigonometry and solving for missing sides in similar triangles.

• Reference angles and coterminal angles are essential for evaluating trigonometric functions for non-standard angles.

• Unit circle values are used to determine exact trigonometric values for common angles.