Trig Exam 1

Complementary angles sum

$90^{\circ}$ or $\frac{\pi}{2}\,\text{radians}$

Complement of an angle in degrees

$$90^{\circ} - \text{given angle}$$

Complement of an angle in radians

$$\frac{\pi}{2} - \text{given angle}$$

Condition when an angle complement is impossible

When the given angle is greater than $90^{\circ}$ or $\frac{\pi}{2}\,\text{radians}$

Supplementary angles sum

$180^{\circ}$ or $\pi\,\text{radians}$

Supplement of an angle in degrees

$$180^{\circ} - \text{given angle}$$

Supplement of an angle in radians

$$\pi - \text{given angle}$$

Conversion factor from degrees to radians

Multiply by $\frac{\pi}{180^{\circ}}$

Conversion factor from radians to degrees

Multiply by $\frac{180^{\circ}}{\pi}$

Degrees in one complete revolution

$$360^{\circ}$$

Radians in one complete revolution

$$2\pi\,\text{radians}$$

Conversion from revolutions to radians

Multiply the number of revolutions by $2\pi$

Minutes in one degree

$60\,\text{minutes}$ ($1^{\circ} = 60'$)

Seconds in one minute

$60\,\text{seconds}$ ($1' = 60''$)

Steps to convert decimal degrees to DMS

Keep the whole number as degrees; multiply the decimal portion by $60$ to get minutes; multiply the remaining decimal portion by $60$ to get seconds.

Formula to convert DMS to decimal degrees

$$\text{Degrees} + \frac{\text{Minutes}}{60} + \frac{\text{Seconds}}{3600}$$

Arc-length formula

$$s = r\theta$$

Required unit for $\theta$ in the arc-length formula $s = r\theta$

$\theta$ must be measured in radians.

Central angle formula using arc length and radius

$$\theta = \frac{s}{r}$$

Radius formula using arc length and central angle

$$r = \frac{s}{\theta}$$

Angular speed formula

$$\omega = \frac{\theta}{t}$$

Relationship between linear speed and angular speed

$$v = r\omega$$

Angular speed formula using linear speed and radius

$$\omega = \frac{v}{r}$$

Meaning of the mnemonic component SOH

$$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$

Meaning of the mnemonic component CAH

$$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$

Meaning of the mnemonic component TOA

$$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$

Identification of the hypotenuse in a right triangle

The longest side, located directly across from the $90^{\circ}$ angle.

Identification of the opposite side in a right triangle

The side directly across from the selected angle.

Identification of the adjacent side in a right triangle

The non-hypotenuse side touching the selected angle.

Pythagorean theorem

$$a^2 + b^2 = c^2$$

Formula for finding a missing leg of a right triangle

$$\text{leg} = \sqrt{\text{hypotenuse}^2 - \text{known leg}^2}$$

Reciprocal identity for cosecant

$$\csc(\theta) = \frac{1}{\sin(\theta)}$$

Reciprocal identity for secant

$$\sec(\theta) = \frac{1}{\cos(\theta)}$$

Reciprocal identity for cotangent

$$\cot(\theta) = \frac{1}{\tan(\theta)}$$

Cosecant value if $\sin(\theta) = \frac{a}{b}$

$$\csc(\theta) = \frac{b}{a}$$

Secant value if $\cos(\theta) = \frac{a}{b}$

$$\sec(\theta) = \frac{b}{a}$$

Cotangent value if $\tan(\theta) = \frac{a}{b}$

$$\cot(\theta) = \frac{b}{a}$$

Distance formula from origin to point $(x,y)$

$$r = \sqrt{x^2 + y^2}$$

Definition of $\sin(\theta)$ for point $(x,y)$

$$\sin(\theta) = \frac{y}{r}$$

Definition of $\cos(\theta)$ for point $(x,y)$

$$\cos(\theta) = \frac{x}{r}$$

Definition of $\tan(\theta)$ for point $(x,y)$

$$\tan(\theta) = \frac{y}{x}$$

Definition of $\csc(\theta)$ for point $(x,y)$

$$\csc(\theta) = \frac{r}{y}$$

Definition of $\sec(\theta)$ for point $(x,y)$

$$\sec(\theta) = \frac{r}{x}$$

Definition of $\cot(\theta)$ for point $(x,y)$

$$\cot(\theta) = \frac{x}{y}$$

Coordinates of a point on the unit circle

$$(\cos(\theta), \sin(\theta))$$

Positive trigonometric functions in Quadrant I

All six trigonometric functions

Positive trigonometric functions in Quadrant II

Sine and cosecant

Positive trigonometric functions in Quadrant III

Tangent and cotangent

Positive trigonometric functions in Quadrant IV

Cosine and secant

Mnemonic for positive trigonometric functions by quadrant

"All Students Take Calculus": Quadrant I: All, Quadrant II: Sine, Quadrant III: Tangent, Quadrant IV: Cosine

Reference angle formula in Quadrant I

$$\theta' = \theta$$

Reference angle formula in Quadrant II (degrees)

$$\theta' = 180^{\circ} - \theta$$

Reference angle formula in Quadrant III (degrees)

$$\theta' = \theta - 180^{\circ}$$

Reference angle formula in Quadrant IV (degrees)

$$\theta' = 360^{\circ} - \theta$$

Reference angle formula in Quadrant II (radians)

$$\theta' = \pi - \theta$$

Reference angle formula in Quadrant III (radians)

$$\theta' = \theta - \pi$$

Reference angle formula in Quadrant IV (radians)

$$\theta' = 2\pi - \theta$$

Procedure for finding a positive coterminal angle (degrees)

Add or subtract $360^{\circ}$ until the angle is between $0^{\circ}$ and $360^{\circ}$

Procedure for finding a positive coterminal angle (radians)

Add or subtract $2\pi$ until the angle is between $0$ and $2\pi$

Test for an even function

$$f(-x) = f(x)$ me

Test for an odd function

$$f(-x) = -f(x)$ me

First step in solving angle-of-elevation or angle-of-depression problems

Draw a right triangle and label the known sides and angle

Angle of elevation

An angle measured upward from a horizontal line

Angle of depression

An angle measured downward from a horizontal line

First strategy when solving a trigonometric identity

Replace complicated trigonometric functions using identities from the formula sheet

Reason $\tan(t)$ can be negative when using $\tan^2(t) = \sec^2(t) - 1$

Taking the square root yields $\pm\sqrt{\sec^2(t) - 1}$; choose the sign based on the quadrant

Expression for $\tan(t)$ in terms of $\sec(t)$

$\tan(t) = \pm\sqrt{\sec^2(t) - 1}$, with the sign determined by the quadrant

Expression for $\csc(t)$ in terms of $\cot(t)$

$\csc(t) = \pm\sqrt{\cot^2(t) + 1}$, with the sign determined by the quadrant

Calculator mode for degree problems

Degree mode

Calculator mode for radian or $\pi$ problems

Radian mode