Trigonometric functions are powerful tools for understanding angles and relationships in triangles. They're essential in math, physics, and engineering. These functions include sine, cosine, tangent, and their reciprocals: , , and .
Knowing exact values of trig functions for common angles is crucial. We'll explore how to calculate these values, work with reference angles beyond the first quadrant, and understand even and odd trig functions. We'll also dive into fundamental identities and using technology for trig calculations.
Trigonometric Functions
Exact values of trigonometric functions
Top images from around the web for Exact values of trigonometric functions
Trigonometric Functions and the Unit Circle | Boundless Algebra View original
Is this image relevant?
TrigCheatSheet.com: Unit Circle Trigonometry View original
Is this image relevant?
MrAllegretti - Trigonometric Functions - B1 View original
Is this image relevant?
Trigonometric Functions and the Unit Circle | Boundless Algebra View original
Is this image relevant?
TrigCheatSheet.com: Unit Circle Trigonometry View original
Is this image relevant?
1 of 3
Top images from around the web for Exact values of trigonometric functions
Trigonometric Functions and the Unit Circle | Boundless Algebra View original
Is this image relevant?
TrigCheatSheet.com: Unit Circle Trigonometry View original
Is this image relevant?
MrAllegretti - Trigonometric Functions - B1 View original
Is this image relevant?
Trigonometric Functions and the Unit Circle | Boundless Algebra View original
Is this image relevant?
TrigCheatSheet.com: Unit Circle Trigonometry View original
Is this image relevant?
1 of 3
Secant (sec) is the reciprocal of cosine divides 1 by the cosine value of an angle secθ=cosθ1
For angle 0, sec0=cos01=11=1 since cosine of 0 is 1
For angle 6π, sec6π=cos6π1=231=323 since cosine of 6π is 23
Cosecant (csc) is the reciprocal of sine divides 1 by the sine value of an angle cscθ=sinθ1
For angle 2π, csc2π=sin2π1=11=1 since sine of 2π is 1
For angle 3π, csc3π=sin3π1=231=323 since sine of 3π is 23
Tangent (tan) is the ratio of sine to cosine divides the sine value by the cosine value of an angle tanθ=cosθsinθ
For angle 4π, tan4π=cos4πsin4π=2222=1 since both sine and cosine of 4π are 22
For angle 3π, tan3π=cos3πsin3π=2123=3 since sine of 3π is 23 and cosine of 3π is 21
Cotangent (cot) is the reciprocal of tangent divides the cosine value by the sine value of an angle cotθ=tanθ1=sinθcosθ
For angle 4π, cot4π=sin4πcos4π=2222=1 since both cosine and sine of 4π are 22
For angle 6π, cot6π=sin6πcos6π=2123=3 since cosine of 6π is 23 and sine of 6π is 21
These can be visualized and calculated using the
Reference angles beyond first quadrant
Reference angle is the acute angle formed between the terminal side of an angle and the x-axis
To find the reference angle, subtract the given angle from the nearest multiple of 2π or 90∘
For angle 45π, the nearest multiple of 2π is 23π, so the reference angle is 23π−45π=4π
The sign of the trigonometric function value depends on the quadrant of the terminal side
Quadrant I (0 to 2π): All functions are positive
Quadrant II (2π to π): Only sine and cosecant are positive
Quadrant III (π to 23π): Only tangent and cotangent are positive
Quadrant IV (23π to 2π): Only cosine and secant are positive
To evaluate cos(45π), find the reference angle 4π and note that 45π is in Quadrant III where cosine is negative, so cos(45π)=−22
Even vs odd trigonometric functions
Even functions are symmetric about the y-axis satisfies f(−θ)=f(θ)
Cosine and secant are even functions
cos(−θ)=cos(θ) (cosine of the negative angle equals cosine of the positive angle)
sec(−θ)=sec(θ) (secant of the negative angle equals secant of the positive angle)
Odd functions are symmetric about the origin satisfies f(−θ)=−f(θ)
Sine, cosecant, tangent, and cotangent are odd functions
sin(−θ)=−sin(θ) (sine of the negative angle equals the negative of sine of the positive angle)
csc(−θ)=−csc(θ) (cosecant of the negative angle equals the negative of cosecant of the positive angle)
tan(−θ)=−tan(θ) (tangent of the negative angle equals the negative of tangent of the positive angle)
cot(−θ)=−cot(θ) (cotangent of the negative angle equals the negative of cotangent of the positive angle)
Trigonometric Identities and Technology
Fundamental trigonometric identities
relate the square of sine and cosine, tangent and secant, or cotangent and cosecant
sin2θ+cos2θ=1 (square of sine plus square of cosine equals 1)
1+tan2θ=sec2θ (1 plus square of tangent equals square of secant)
1+cot2θ=csc2θ (1 plus square of cotangent equals square of cosecant)
relate sine and cosecant, cosine and secant, or tangent and cotangent
sinθ=cscθ1 (sine equals the reciprocal of cosecant)
cosθ=secθ1 (cosine equals the reciprocal of secant)
tanθ=cotθ1 (tangent equals the reciprocal of cotangent)
Quotient identities express tangent as the ratio of sine to cosine and cotangent as the ratio of cosine to sine
tanθ=cosθsinθ (tangent equals sine divided by cosine)
cotθ=sinθcosθ (cotangent equals cosine divided by sine)
Technology in trigonometric evaluation
Most scientific calculators have buttons for trigonometric functions
Make sure the calculator is in the correct mode (degrees or radians)
For angle 150∘, switch calculator to degree mode, press the "sin" button, enter 150, then press "=" to get sin(150∘)=21
For angle 65π, switch calculator to radian mode, press the "tan" button, enter 65π, then press "=" to get tan(65π)=−3
Spreadsheet software (Excel, Google Sheets) also have trigonometric functions
Use the
SIN()
,
COS()
,
TAN()
,
CSC()
,
SEC()
, and
COT()
functions
For angle 3π, enter
=SIN(PI()/3)
into a cell to get sin(3π)=23
Programming languages (Python, JavaScript) have built-in trigonometric functions in their math libraries
In Python, use
math.sin()
,
math.cos()
,
math.tan()
,
math.csc()
,
math.sec()
, and
math.cot()
For angle 32π, use
math.cos(2*math.pi/3)
to get cos(32π)=−21
Advanced Trigonometric Concepts
is an alternative way to express angles, where one radian is the angle subtended by an arc length equal to the radius of the circle
, such as arcsin, arccos, and arctan, allow us to find angles given trigonometric ratios
helps visualize their periodic nature and key features like amplitude, , and phase shifts