Precalculus Trigonometry: Identities, Formulas, and Applications Study Guide

Trigonometric Functions and Identities

Evaluating Trigonometric Functions

Trigonometric functions describe relationships between the angles and sides of triangles, and are fundamental in Precalculus. Problems often require finding the value of one trigonometric function given another, or using identities to simplify expressions.

• Key Trigonometric Functions: Sine ($\sin$), Cosine ($\cos$), Tangent ($\tan$), Secant ($\sec$), Cosecant ($\csc$), Cotangent ($\cot$).

• Reciprocal Relationships: $\csc \theta = \frac{1}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, $\cot \theta = \frac{1}{\tan \theta}$.

• Pythagorean Identities: $\sin^2 \theta + \cos^2 \theta = 1$, $1 + \tan^2 \theta = \sec^2 \theta$, $1 + \cot^2 \theta = \csc^2 \theta$.

• Example: If $\cos \theta = \frac{8}{17}$ and $\frac{\pi}{2} < \theta < 2\pi$, find $\sin(2\theta)$ using the double-angle formula.

Double-Angle and Half-Angle Formulas

These formulas allow the calculation of trigonometric functions of multiple or half angles in terms of functions of the original angle.

• Double-Angle Formulas:

  • $\sin(2\theta) = 2\sin \theta \cos \theta$

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

• Half-Angle Formulas:

  • $\sin^2 \frac{\theta}{2} = \frac{1 - \cos \theta}{2}$

  • $\cos^2 \frac{\theta}{2} = \frac{1 + \cos \theta}{2}$

  • $\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}$

  • $\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}$

• Example: Find $\sin(165^\circ)$ using the half-angle formula.

Sum-to-Product and Product-to-Sum Formulas

These identities convert sums or differences of sines and cosines into products, and vice versa, which is useful for simplifying expressions and solving equations.

• Product-to-Sum:

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

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

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

• Sum-to-Product:

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

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

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

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

• Example: Express $\sin(50^\circ)\sin(40^\circ)$ as a sum.

Solving Triangles

Law of Sines and Law of Cosines

These laws are used to solve for unknown sides or angles in non-right triangles.

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

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

• Applications: Used when given two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA), or all three sides (SSS).

• Example: Given $a = 8$, $b = 7$, $\gamma = 87^\circ$, find $c$, $\alpha$, and $\beta$.

Area of a Triangle

The area of a triangle can be found using trigonometric formulas, especially when two sides a known identities.

• Example: $\sin(15^\circ)\sin(40^\circ) = ?$

• Strategy: Use product-to-sum or sum-to-product formulas to rewrite the expression.

Applications of Trigonometry

• Heron's Formula: $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$

• Example: Find the area of a triangle with sides 6 ft, 8 ft, and included angle $125^\circ$.

Trigonometric Equations and Identities

Angle of Elevation and Depressionnd the included angle are known.

• Formula: $\text{Area} = \frac{1}{2}ab\sin C$

Completing and Verifying Identities

Trigonometric identities are equations that are true for all values of the variable where both sides are defined. Completing or verifying identities involves transforming one side of the equation to match the other using

Problems involving the angle of elevation or depression use trigonometric ratios to find heights or distances that are not directly measurable.

• Key Concept: The angle of elevation is the angle above horizontal from the observer's eye to an object.

• Example: Estimating the height of a tree using two different angles of elevation from two points a known distance apart.

Navigation ry Table: Key Trigonometric Formulas

Formula Type	Formula	Application
Pythagorean Identity	$\sin^2 \theta + \cos^2 \theta = 1$	Relating sine and cosine
Double-Angle (Sine)	$\sin(2\theta) = 2\sin \theta \cos \theta$	Finding sine of double an angle
and Bearings Bearings are used in navigation to describe direction. Trigonometry is used to solve problems involving distances and directions between points. Bearing: The direction or path along which something moves or along which it lies. Example: Calculating the distance a ship has sailed using bearings from two points. Summa Double-Angle (Cosine)	$\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$	Finding cosine of double an angle

Half-Angle (Sine)	$\sin^2 \frac{\theta}{2} = \frac{1 - \cos \theta}{2}$	Finding sine of half an angle
Law of Sines	$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$	Solving triangles
Law of Cosines	$c^2 = a^2 + b^2 - 2ab\cos C$	Solving triangles
Area of Triangle	$\text{Area} = \frac{1}{2}ab\sin C$	Finding area with two sides and included angle

Practice and Application

• Apply identities to simplify trigonometric expressions.

• Use the Law of Sines and Law of Cosines to solve for unknown sides and angles in triangles.

• Calculate areas of triangles using trigonometric formulas.

• Solve real-world problems involving angles of elevation, bearings, and navigation.

Additional info: This study guide is based on a set of Precalculus trigonometry multiple-choice questions covering identities, formulas, triangle solving, and applications. All key formulas and concepts have been expanded for clarity and completeness.