1.3 Trigonometric Functions - Introduction to Calculus

Trigonometric Functions

Let $O$ be the origin and $P(x, y)$ be a point on the unit circle so that the radius $OP$ forms an angle of $\theta$ radians with respect to the positive x-axis.

Follow P on the unit circle. Blue measures cosine, red measures sine, and the green tangent line at x = 1 gives tan(θ) when it is defined.

θ (degrees)45

\displaystyle \theta=45^\circ=0.785\ \text{rad}

\displaystyle \cos(\theta)=0.707

\displaystyle \sin(\theta)=0.707

\displaystyle \tan(\theta)=1

Try 0°, 45°, and 90°. Compare the projections and tangent.

THEOREM

sin / cos

\begin{aligned} & x = \cos(\theta) \\[0.65em] & y = \sin(\theta) \end{aligned}

Here are the three most common trigonometric functions and their reciprocals.

\displaystyle y=\sin(x)

Property	$y=\sin(\theta)$
Domain	$\theta\in\mathbb{R}$
Range	$-1\le\sin(\theta)\le1$
Zeros	$\theta=k\pi$

Here $k\in\mathbb{Z}$

\displaystyle y=\cos(x)

Property	$y=\cos(\theta)$
Domain	$\theta\in\mathbb{R}$
Range	$-1\le\cos(\theta)\le1$
Zeros	$\theta=(2k+1)\pi/2$

Here $k\in\mathbb{Z}$

\displaystyle y=\tan(x)

Property	$y=\tan(\theta)$
Domain	$\theta\in\mathbb{R},\ \theta\ne(2k+1)\pi/2$
Range	$\tan(\theta)\in\mathbb{R}$
Zeros	$\theta=k\pi$

Here $k\in\mathbb{Z}$

QUESTION

What happens to the period when

\displaystyle y=\sin(x)

becomes

\displaystyle y=\sin(2x)

Explore $\displaystyle y=A\,\sin(Bx+C)+D$ . Change one parameter at a time and compare the colored curve with the original. Dashed lines mark vertical asymptotes.

Function

\displaystyle y=\sin(x)\quad\text{(reference)}

\displaystyle y=\sin(x)

A · vertical scale1B · frequency factor1C · phase (radians)0D · vertical shift0Input x (radians)0

\displaystyle f(0)=0

Amplitude = 1.0 · Period = 6.283 · Horizontal shift = 0.000

Compare sine with tangent. Double B and observe the period; move x toward a tangent asymptote and watch the output.

Reciprocal Trigonometric Functions

$\csc(\theta)=1/\sin(\theta)$ , $\sec(\theta)=1/\cos(\theta)$ , and $\cot(\theta)=1/\tan(\theta)$ wherever the expressions are defined.

\displaystyle y=\csc(x)

Property	$y=\csc(\theta)$
Domain	$\theta\in\mathbb{R},\ \theta\ne k\pi$
Range	$\csc(\theta)\le-1$ or $\csc(\theta)\ge1$
Zeros	Never

Here $k\in\mathbb{Z}$

\displaystyle y=\sec(x)

Property	$y=\sec(\theta)$
Domain	$\theta\in\mathbb{R},\ \theta\ne(2k+1)\pi/2$
Range	$\sec(\theta)\le-1$ or $\sec(\theta)\ge1$
Zeros	Never

Here $k\in\mathbb{Z}$

\displaystyle y=\cot(x)

Property	$y=\cot(\theta)$
Domain	$\theta\in\mathbb{R},\ \theta\ne k\pi$
Range	$\cot(\theta)\in\mathbb{R}$
Zeros	$\theta=(2k+1)\pi/2$

Here $k\in\mathbb{Z}$

QUESTION

Why is

\displaystyle \csc(0)

undefined?

Explore $\displaystyle y=A\,\csc(Bx+C)+D$ . Change one parameter at a time and compare the colored curve with the original. Dashed lines mark vertical asymptotes.

Function

\displaystyle y=\csc(x)\quad\text{(reference)}

\displaystyle y=\csc(x)

A · vertical scale1B · frequency factor1C · phase (radians)0D · vertical shift0Input x (radians)0

\displaystyle f(0)=\text{undefined}

Vertical scale = 1.0 · Period = 6.283 · Horizontal shift = 0.000

This input is excluded from the domain because the denominator is zero.

Compare the gaps in each graph with the zeros of its denominator. Move C to shift the excluded inputs; change B to bring the branches closer together.

Trigonometric Functions

Let $O$ be the origin and $P(x, y)$ be a point on the unit circle so that the radius $OP$ forms an angle of $\theta$ radians with respect to the positive x-axis.

Follow P on the unit circle. Blue measures cosine, red measures sine, and the green tangent line at x = 1 gives tan(θ) when it is defined.

θ (degrees)45

\displaystyle \theta=45^\circ=0.785\ \text{rad}

\displaystyle \cos(\theta)=0.707

\displaystyle \sin(\theta)=0.707

\displaystyle \tan(\theta)=1

Try 0°, 45°, and 90°. Compare the projections and tangent.

THEOREM

sin / cos

\begin{aligned} & x = \cos(\theta) \\[0.65em] & y = \sin(\theta) \end{aligned}

Here are the three most common trigonometric functions and their reciprocals.

\displaystyle y=\sin(x)

Property	$y=\sin(\theta)$
Domain	$\theta\in\mathbb{R}$
Range	$-1\le\sin(\theta)\le1$
Zeros	$\theta=k\pi$

Here $k\in\mathbb{Z}$

\displaystyle y=\cos(x)

Property	$y=\cos(\theta)$
Domain	$\theta\in\mathbb{R}$
Range	$-1\le\cos(\theta)\le1$
Zeros	$\theta=(2k+1)\pi/2$

Here $k\in\mathbb{Z}$

\displaystyle y=\tan(x)

Property	$y=\tan(\theta)$
Domain	$\theta\in\mathbb{R},\ \theta\ne(2k+1)\pi/2$
Range	$\tan(\theta)\in\mathbb{R}$
Zeros	$\theta=k\pi$

Here $k\in\mathbb{Z}$

QUESTION

What happens to the period when

\displaystyle y=\sin(x)

becomes

\displaystyle y=\sin(2x)

Explore $\displaystyle y=A\,\sin(Bx+C)+D$ . Change one parameter at a time and compare the colored curve with the original. Dashed lines mark vertical asymptotes.

Function

\displaystyle y=\sin(x)\quad\text{(reference)}

\displaystyle y=\sin(x)

A · vertical scale1B · frequency factor1C · phase (radians)0D · vertical shift0Input x (radians)0

\displaystyle f(0)=0

Amplitude = 1.0 · Period = 6.283 · Horizontal shift = 0.000

Compare sine with tangent. Double B and observe the period; move x toward a tangent asymptote and watch the output.

Reciprocal Trigonometric Functions

$\csc(\theta)=1/\sin(\theta)$ , $\sec(\theta)=1/\cos(\theta)$ , and $\cot(\theta)=1/\tan(\theta)$ wherever the expressions are defined.

\displaystyle y=\csc(x)

Property	$y=\csc(\theta)$
Domain	$\theta\in\mathbb{R},\ \theta\ne k\pi$
Range	$\csc(\theta)\le-1$ or $\csc(\theta)\ge1$
Zeros	Never

Here $k\in\mathbb{Z}$

\displaystyle y=\sec(x)

Property	$y=\sec(\theta)$
Domain	$\theta\in\mathbb{R},\ \theta\ne(2k+1)\pi/2$
Range	$\sec(\theta)\le-1$ or $\sec(\theta)\ge1$
Zeros	Never

Here $k\in\mathbb{Z}$

\displaystyle y=\cot(x)

Property	$y=\cot(\theta)$
Domain	$\theta\in\mathbb{R},\ \theta\ne k\pi$
Range	$\cot(\theta)\in\mathbb{R}$
Zeros	$\theta=(2k+1)\pi/2$

Here $k\in\mathbb{Z}$

QUESTION

Why is

\displaystyle \csc(0)

undefined?

Explore $\displaystyle y=A\,\csc(Bx+C)+D$ . Change one parameter at a time and compare the colored curve with the original. Dashed lines mark vertical asymptotes.

Function

\displaystyle y=\csc(x)\quad\text{(reference)}

\displaystyle y=\csc(x)

A · vertical scale1B · frequency factor1C · phase (radians)0D · vertical shift0Input x (radians)0

\displaystyle f(0)=\text{undefined}

Vertical scale = 1.0 · Period = 6.283 · Horizontal shift = 0.000

This input is excluded from the domain because the denominator is zero.

Compare the gaps in each graph with the zeros of its denominator. Move C to shift the excluded inputs; change B to bring the branches closer together.

Trigonometric Functions

Trigonometric Functions

Read sine, cosine, and tangent from the unit circle

sin / cos

KNOWLEDGE QUIZ

Transform sine, cosine, and tangent

Reciprocal Trigonometric Functions

KNOWLEDGE QUIZ

Explore cosecant, secant, and cotangent

Trigonometric Functions

Trigonometric Functions

Read sine, cosine, and tangent from the unit circle

sin / cos

KNOWLEDGE QUIZ

Transform sine, cosine, and tangent

Reciprocal Trigonometric Functions

KNOWLEDGE QUIZ

Explore cosecant, secant, and cotangent