are powerful tools for analyzing angles and ratios in triangles. They include sine, cosine, , and their reciprocals: , , and . These functions have exact values for common angles and can be evaluated using reference angles.
Understanding the properties of trigonometric functions is crucial. Some are even, others odd. Fundamental identities relate these functions to each other. Knowing how to use a calculator for trig functions and recognizing their characteristics like periodicity and amplitude is essential for solving real-world problems.
Trigonometric Functions
Exact values of trigonometric functions
Secant (sec)
Reciprocal of cosine sec(θ)=cos(θ)1
sec(6π)=32 hypotenuse divided by adjacent side in a 30-60-90 triangle
sec(4π)=2 hypotenuse divided by adjacent side in a 45-45-90 triangle
sec(3π)=2 hypotenuse divided by adjacent side in a 30-60-90 triangle
Cosecant (csc)
Reciprocal of sine csc(θ)=sin(θ)1
csc(6π)=2 hypotenuse divided by opposite side in a 30-60-90 triangle
csc(4π)=2 hypotenuse divided by opposite side in a 45-45-90 triangle
csc(3π)=32 hypotenuse divided by opposite side in a 30-60-90 triangle
Tangent (tan)
Ratio of opposite side to adjacent side tan(θ)=adjacentopposite
tan(6π)=33 opposite side divided by adjacent side in a 30-60-90 triangle
tan(4π)=1 opposite side equals adjacent side in a 45-45-90 triangle
tan(3π)=3 opposite side divided by adjacent side in a 30-60-90 triangle
Cotangent (cot)
Reciprocal of tangent cot(θ)=tan(θ)1=oppositeadjacent
cot(6π)=3 adjacent side divided by opposite side in a 30-60-90 triangle
cot(4π)=1 adjacent side equals opposite side in a 45-45-90 triangle
cot(3π)=33 adjacent side divided by opposite side in a 30-60-90 triangle
Reference angles for function evaluation
Reference angles
Acute angles between 0 and 2π radians or 0° and 90°
Simplify evaluation of trigonometric functions by relating angles in other quadrants to reference angles
Used to determine function values on the unit circle
Secant
\sec(\theta) = \sec(\text{[reference angle](https://www.fiveableKeyTerm:reference_angle)}) for angles in quadrants I (0≤θ≤2π) or IV (23π≤θ≤2π)
sec(θ)=−sec(reference angle) for angles in quadrants II (2π≤θ≤π) or III (π≤θ≤23π)
Tangent
tan(θ)=tan(reference angle) for angles in quadrants I (0≤θ≤2π) or III (π≤θ≤23π)
tan(θ)=−tan(reference angle) for angles in quadrants II (2π≤θ≤π) or IV (23π≤θ≤2π)
Cotangent
cot(θ)=cot(reference angle) for angles in quadrants I (0≤θ≤2π) or III (π≤θ≤23π)
cot(θ)=−cot(reference angle) for angles in quadrants II (2π≤θ≤π) or IV (23π≤θ≤2π)
Properties of Trigonometric Functions
Even vs odd trigonometric functions
Even functions
Symmetric about the y-axis f(−x)=f(x)
Cosine: cos(−θ)=cos(θ) (reflection over y-axis yields same value)
Secant: sec(−θ)=sec(θ) (reflection over y-axis yields same value)
Odd functions
Symmetric about the origin f(−x)=−f(x)
Sine: sin(−θ)=−sin(θ) (reflection over origin yields negated value)
Tangent: tan(−θ)=−tan(θ) (reflection over origin yields negated value)
Cosecant: csc(−θ)=−csc(θ) (reflection over origin yields negated value)
Cotangent: cot(−θ)=−cot(θ) (reflection over origin yields negated value)
Fundamental trigonometric identities
Reciprocal identities relate trigonometric functions to their reciprocals
sin(θ)=csc(θ)1 sine is the reciprocal of cosecant
cos(θ)=sec(θ)1 cosine is the reciprocal of secant
tan(θ)=cot(θ)1 tangent is the reciprocal of cotangent
Pythagorean identities based on the Pythagorean theorem a2+b2=c2
sin2(θ)+cos2(θ)=1 square of sine plus square of cosine equals 1
1+tan2(θ)=sec2(θ) 1 plus square of tangent equals square of secant
1+cot2(θ)=csc2(θ) 1 plus square of cotangent equals square of cosecant
Quotient identities express tangent and cotangent as ratios of sine and cosine
tan(θ)=cos(θ)sin(θ) tangent is sine divided by cosine
cot(θ)=sin(θ)cos(θ) cotangent is cosine divided by sine
Calculator use in trigonometry
Set calculator to correct angle mode
Degrees for angles measured in degrees
Radians for angles measured in radians (π units)
Use appropriate trigonometric function buttons
sin
,
cos
,
tan
for sine, cosine, tangent
sec
,
csc
,
cot
for secant, cosecant, cotangent (if available)
Enter angle value and press corresponding function button to evaluate
sin(30°)=0.5 in degree mode
cos(3π)≈0.5 in radian mode
Be aware of calculator's limitations
Very large angles may yield inaccurate results due to rounding errors
Very small angles close to 0 may be treated as 0 by calculator
Some calculators have inverse trigonometric function buttons for finding angles
Characteristics of Trigonometric Functions
Periodic functions: Trigonometric functions repeat their values at regular intervals
Amplitude: The maximum distance between the midline and the graph of a trigonometric function
Domain and range: The set of possible input (x) and output (y) values for trigonometric functions
Phase shift: Horizontal translation of a trigonometric function's graph
Vertical shift: Vertical translation of a trigonometric function's graph
Key Terms to Review (8)
Cosecant: The cosecant function, denoted as $\csc(\theta)$, is the reciprocal of the sine function. It is defined as $\csc(\theta) = \frac{1}{\sin(\theta)}$ where $\sin(\theta) \neq 0$.
Cotangent: Cotangent is a trigonometric function defined as the reciprocal of the tangent function. It can be expressed as $\cot(\theta) = \frac{1}{\tan(\theta)}$ or $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$.
Period: The period of a trigonometric function is the interval over which it completes one full cycle and starts to repeat. For sine and cosine functions, the period is $2\pi$.
Pythagorean Identity: The Pythagorean Identity is a fundamental relation in trigonometry that states $\sin^2(\theta) + \cos^2(\theta) = 1$. It holds for any angle $\theta$ and is derived from the Pythagorean Theorem applied to the unit circle.
Reference angle: A reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis. It is always between $0^{\circ}$ and $90^{\circ}$ inclusive.
Secant: A secant function, denoted as $\sec(\theta)$, is the reciprocal of the cosine function. It is defined as $\sec(\theta) = \frac{1}{\cos(\theta)}$.
Tangent: A tangent is a trigonometric function represented as $\tan(\theta)$, which is the ratio of the sine and cosine of an angle: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. It is undefined when $\cos(\theta) = 0$.
Trigonometric functions: Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are fundamental in studying periodic phenomena and are defined using the unit circle.