Right Triangle Trigonometry and Angles

9.1 Right Triangle Trigonometry

Introduction

Right triangle trigonometry involves the study of ratios of side lengths of right triangles related to their acute angles. These ratios define six trigonometric functions: sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).

Definitions of Trigonometric Functions

Consider an acute angle θ\theta in a right triangle:

  • Sine (sin): The ratio of the length of the opposite side to the length of the hypotenuse.sinθ=opphypsin \theta = \frac{opp}{hyp}

  • Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse. cosθ=adjhypcos \theta = \frac{adj}{hyp}

  • Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side. tanθ=oppadjtan \theta = \frac{opp}{adj}

  • Cotangent (cot): The ratio of the length of the adjacent side to the length of the opposite side. It is the reciprocal of the tangent function.cotθ=adjoppcot \theta = \frac{adj}{opp}

  • Secant (sec): The ratio of the length of the hypotenuse to the length of the adjacent side. It is the reciprocal of the cosine function.secθ=hypadjsec \theta = \frac{hyp}{adj}

  • Cosecant (csc): The ratio of the length of the hypotenuse to the length of the opposite side. It is the reciprocal of the sine function.cscθ=hypoppcsc \theta = \frac{hyp}{opp}

Reciprocal Relationships

The trigonometric functions have reciprocal relationships:

  • csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}

  • sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}

  • cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta}

Evaluating Trigonometric Functions

To evaluate the six trigonometric functions of an angle θ\theta:

  1. Identify the lengths of the opposite side, adjacent side, and hypotenuse with respect to the angle θ\theta.

  2. Use the definitions of the trigonometric functions to calculate their values.

Example

Given a right triangle with an acute angle theta\\theta, where the opposite side is 5 and the hypotenuse is 13. To find the adjacent side:

adj=132−52=169−25=144=12{adj = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12}

Then, the six trigonometric functions are:

  • sin⁡θ=513\sin \theta = \frac{5}{13}

  • cos⁡θ=1213\cos \theta = \frac{12}{13}

  • tan⁡θ=512\tan \theta = \frac{5}{12}

  • cot⁡θ=125\cot \theta = \frac{12}{5}

  • sec⁡θ=1312\sec \theta = \frac{13}{12}

  • csc⁡θ=135\csc \theta = \frac{13}{5}

Trigonometric Values for Special Angles

Specific angles such as 30°, 45°, and 60° (or their radian equivalents) have particular trigonometric values that can be derived from special triangles.

30° - 60° - 90° Triangle

In a 30-60-90 triangle, if the side opposite the 30° angle is 1, then the side opposite the 60° angle is 3\sqrt{3} ,and the hypotenuse is 2.

  • sin⁡30∘=12\sin 30^\circ = \frac{1}{2}

  • cos⁡30∘=32\cos 30^\circ = \frac{\sqrt{3}}{2}

  • tan⁡30∘=13=33\tan 30^\circ = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}

  • sin⁡60∘=32\sin 60^\circ = \frac{\sqrt{3}}{2}

  • cos⁡60∘=12\cos 60^\circ = \frac{1}{2}

  • tan⁡60∘=3\tan 60^\circ = \sqrt{3}

45° - 45° - 90° Triangle

In a 45-45-90 triangle, if both legs are 1, then the hypotenuse is 2\sqrt{2}.

  • sin⁡45∘=12=22\sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}

  • cos⁡45∘=12=22\cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}

  • tan⁡45∘=1\tan 45^\circ = 1

Solving Right Triangles

Solving a right triangle involves finding all unknown side lengths and angle measures. Given some information (e.g., two side lengths or one side length and an acute angle), trigonometric functions can be used to find the remaining information.

Using a Calculator

Calculators are used to find trigonometric function values for angles that are not special angles.

Real-Life Problems

Right triangle trigonometry is applied in various real-life scenarios, such as finding heights, distances, and angles of elevation or depression.

9.2 Angles and Radian Measure

Radian Measure

Radian measure provides an alternative way to measure angles. One radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. A full circle (360°) is equal to 2π2\pi radians.

Conversion Between Degrees and Radians

  • To convert degrees to radians, multiply by π180∘\frac{\pi}{180^\circ}.

  • To convert radians to degrees, multiply by 180∘π\frac{180^\circ}{\pi}.

Angles in Standard Position

An angle is in standard position when its vertex is at the origin and its initial side lies on the positive x-axis.

Coterminal Angles

Coterminal angles are angles in standard position that have the same terminal side. Coterminal angles can be found by adding or subtracting multiples of 360° (or 2π2\pi radians).

Arc Length

The arc length (s) of a sector of a circle is given by the formula:

s=rθ{s = r\theta}

where r is the radius of the circle and θ\theta is the central angle in radians.

Area of a Sector

The area (A) of a sector of a circle is given by the formula:

A=12r2θ{A = \frac{1}{2}r^2\theta}

where r is the radius of the circle and θ\theta is the central angle in radians.

9.3 - Trigonometric Functions of Any Angle

General Definitions of Trigonometric Functions

Given an angle theta\\theta in standard position and a point (x,y)(x, y) on the terminal side of the angle that intersects the circle x2+y2=r2x^2 + y^2 = r^2:

  • sin⁡θ=yr\sin \theta = \frac{y}{r}

  • cos⁡θ=xr\cos \theta = \frac{x}{r}

  • tan⁡θ=yx\tan \theta = \frac{y}{x}

  • csc⁡θ=ry\csc \theta = \frac{r}{y}

  • sec⁡θ=rx\sec \theta = \frac{r}{x}

  • cot⁡θ=xy\cot \theta = \frac{x}{y}

The Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin. On the unit circle, the coordinates of the point where the terminal side of an angle theta\\theta intersects the circle are (cos⁡θ,sin⁡θ)(\cos \theta, \sin \theta).

Reference Angles

The reference angle theta′\\theta' for an angle theta\\theta in standard position is the acute angle formed by the terminal side of theta\\theta and the x-axis.

Finding Reference Angles

The reference angle depends on the quadrant in which the terminal side of theta\\theta lies:

  • Quadrant II: θ′=180∘−θ\theta' = 180^\circ - \theta

  • Quadrant III: θ′=θ−180∘\theta' = \theta - 180^\circ

  • Quadrant IV: θ′=360∘−θ\theta' = 360^\circ - \theta

Evaluating Trigonometric Functions Using Reference Angles

  1. Find the reference angle theta′\\theta'.

  2. Evaluate the trigonometric function for theta′\\theta'.

  3. Determine the sign of the trigonometric function value based on the quadrant in which theta\\theta lies.

9.4 Graphing Sine and Cosine Functions

Characteristics of Sine and Cosine Functions

The general forms of the sine and cosine functions:

  • y=asin⁡(bx)y = a \sin(bx)

  • y=acos⁡(bx)y = a \cos(bx)

Where:

  • aa is the amplitude.

  • 2πb\frac{2\pi}{b} is the period.

Amplitude, Period, and Midline

  • Amplitude: The distance from the midline to the maximum or minimum value of the function.

  • Period: The length of one complete cycle of the function.

  • Midline: The horizontal line that runs midway between the maximum and minimum values of the function.

9.5 Graphing Other Trigonometric Functions

Tangent Function

The tangent function is defined as y=tan⁡xy = \tan x

Cotangent Function

The cotangent function is defined as y=cot⁡xy = \cot x

The period of both tangent and cotangent functions is π\pi.

Asymptotes

Tangent and cotangent have vertical asymptotes at values where they are undefined.

Secant and Cosecant Functions

The secant function is defined as y=sec⁡x=1cos⁡xy = \sec x = \frac{1}{\cos x}

The cosecant function is defined as y=csc⁡x=1sin⁡xy = \csc x = \frac{1}{\sin x}

9.7 Trigonometric Identities

Fundamental Trigonometric Identities

  • Reciprocal Identities:

    • csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}

    • sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}

    • cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta}

  • Quotient Identities:

    • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

    • cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta}

  • Pythagorean Identities:

    • sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

    • 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta

    • 1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta

Verifying Trigonometric Identities

To verify an identity, manipulate one side of the equation until it is identical to the other side, using known identities and algebraic manipulations.

9.8 Using Sum and Difference Formulas

Sum and Difference Formulas

  • sin⁡(a+b)=sin⁡acos⁡b+cos⁡asin⁡b\sin(a + b) = \sin a \cos b + \cos a \sin b

  • sin⁡(a−b)=sin⁡acos⁡b−cos⁡asin⁡b\sin(a - b) = \sin a \cos b - \cos a \sin b

  • cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b\cos(a + b) = \cos a \cos b - \sin a \sin b

  • cos⁡(a−b)=cos⁡acos⁡b+sin⁡asin⁡b\cos(a - b) = \cos a \cos b + \sin a \sin b

  • tan⁡(a+b)=tan⁡a+tan⁡b1−tan⁡atan⁡b\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}

  • tan⁡(a−b)=tan⁡a−tan⁡b1+tan⁡atan⁡b\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}

Applications

These formulas are valuable for evaluating trigonometric functions of angles that can be expressed as the sum or difference of special angles and for simplifying trigonometric expressions.