Precalculus Study Notes: Trigonometric Functions, Angles, and Right Triangle Applications

Trigonometric Angles and Unit Conversions

Converting Between Radians, Revolutions, and Degrees

Angles can be measured in degrees, radians, or revolutions. Understanding how to convert between these units is essential in trigonometry.

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

• Degrees to Revolutions: $\text{Revolutions} = \frac{\text{Degrees}}{360}$

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

• Radians to Revolutions: $\text{Revolutions} = \frac{\text{Radians}}{2\pi}$

• Degrees to DMS (Degrees, Minutes, Seconds): Convert the decimal part of degrees to minutes by multiplying by 60, then the decimal part of minutes to seconds by multiplying by 60.

Example: Convert $62.5^\circ$ to radians:

• $62.5^\circ \times \frac{\pi}{180} = 1.09$ radians

Exact Values of Trigonometric Functions

Evaluating Trigonometric Functions for Special Angles

Some trigonometric functions have exact values for special angles, which are commonly used in precalculus.

• cos(30°): $\cos 30^\circ = \frac{\sqrt{3}}{2}$

• tan(60°): $\tan 60^\circ = \sqrt{3}$

• csc(90°): $\csc 90^\circ = 1$

Key Terms:

• Primary trigonometric functions: sine, cosine, tangent

• Secondary trigonometric functions: cosecant, secant, cotangent

Calculator Use and Trigonometric Function Evaluation

Evaluating Trigonometric Functions Using Calculators

Calculators can be used in DEG (degree) or RAD (radian) mode to evaluate trigonometric functions. Secondary functions should be expressed in terms of primary functions before calculation.

• Example: $\sin 14.7^\circ = 0.254$

• Example: $\tan 92.8^\circ = -20.4$

• Example: $\sec 211.5^\circ = -1.91$

In RAD mode, similar calculations apply, but the input is in radians.

• Example: $\cos 2.129 = -0.530$

• Example: $\sec 5.875 = 1.09$

• Example: $\cot 4.112 = 0.685$

Solving for Angles Given Trigonometric Values

Finding Angles from Trigonometric Values

To find the angle $\theta$ given a trigonometric value, use the inverse function and consider the quadrant(s) where the function is positive or negative.

• Example: If $\cos \theta = 0.810$, $\theta = \cos^{-1}(0.810) = 35.9^\circ$. Since cosine is positive in Quadrants I and IV, the other solution is $360^\circ - 35.9^\circ = 324.1^\circ$.

• Example: If $\sec \theta = -1.80$, $\theta = \sec^{-1}(-1.80) = \cos^{-1}(-1/1.80) = 123^\circ$. Since secant is negative in Quadrants II and III, the other solution is $360^\circ - 123^\circ = 236.3^\circ$.

Solving Right Triangles

Using Trigonometric Ratios to Solve Right Triangles

Given some sides or angles of a right triangle, use trigonometric ratios to find unknown values.

• 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: Given $A = 36.5^\circ$, $c = 7.18$:

• Find $B = 90^\circ - 36.5^\circ = 53.5^\circ$

• Find $a = c \sin A = 7.18 \sin 36.5^\circ = 4.27$

• Find $b = c \cos A = 7.18 \cos 36.5^\circ = 5.77$

Applications: Angle of Elevation

Solving Real-World Problems Using Trigonometry

Trigonometric functions can be used to solve problems involving heights and distances, such as finding the height of a building given the angle of elevation and the distance from the building.

• Example: From a point 125 ft from a building, the angle of elevation to the top is $62.6^\circ$. The height $h$ is:

$h = 125 \tan 62.6^\circ = 240 \text{ ft}$

Summary Table: Trigonometric Function Relationships

Function	Definition	Quadrants (+/-)
sin	$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$	+ in I, II; - in III, IV
cos	$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$	+ in I, IV; - in II, III
tan	$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$	+ in I, III; - in II, IV
csc	$\csc \theta = \frac{1}{\sin \theta}$	Same as sin
sec	$\sec \theta = \frac{1}{\cos \theta}$	Same as cos
cot	$\cot \theta = \frac{1}{\tan \theta}$	Same as tan

Additional info: Some explanations and examples were expanded for clarity and completeness, including quadrant analysis and step-by-step triangle solutions.