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Grade 11

Intro to sin(x), cos(x) and tan(x) graphs

Lesson

The definitions of the trigonometric functions $\sin$sin, $\cos$cos and $\tan$tan that we've seen use the ratios of side lengths of a right-angled triangle. More specifically, we call this the right-angled triangle definition of the trigonometric functions, but there are other methods to define these functions more broadly.

Right-angled triangle definition

For a right-angled triangle, where $\theta$θ is the measure for one of the angles (excluding the right angle), we have that:

$\sin\theta$sinθ $=$= $\frac{\text{opposite }}{\text{hypotenuse }}$opposite hypotenuse
$\cos\theta$cosθ $=$= $\frac{\text{adjacent }}{\text{hypotenuse }}$adjacent hypotenuse
$\tan\theta$tanθ $=$= $\frac{\text{opposite }}{\text{adjacent }}$opposite adjacent

 

Now consider a right-angled triangle, with hypotenuse that has a length of one unit with a vertex centred at the origin. We can construct a unit circle around the triangle as shown below.

A right-angled triangle inscribed in the unit circle.

 

The point indicated on the circle has coordinates $\left(\cos\theta,\sin\theta\right)$(cosθ,sinθ) using the right-angled triangle definition of $\cos$cos and $\sin$sin. Unfortunately, this definition is limited to angles with measures in the range of $0^\circ\le\theta\le90^\circ$0°θ90°. However, more broadly, we can use the unit circle to define $\cos$cos and $\sin$sin for angles with any measure. We call this the unit circle definition. In this definition, the value of these functions will be the $x$x- and $y$y-values of a point on the unit circle after having rotated by an angle of measure $\theta$θ in the counterclockwise direction as shown below. If $\theta$θ is negative then the point is rotated in the clockwise direction.

Definition of $\cos$cos and $\sin$sin can extend beyond $0^\circ\le\theta\le90^\circ$0°θ90°.

 

As we move through different values of $\theta$θ the value of $\cos\theta$cosθ and $\sin\theta$sinθ move accordingly between $-1$1 and $1$1. If we plot the values of $\cos\theta$cosθ and $\sin\theta$sinθ according to different values of $\theta$θ on the unit circle, we get the following graphs:

$y=\cos\theta$y=cosθ

 

$y=\sin\theta$y=sinθ

 

As in the right-angled triangle definition, we still define $\tan\theta$tanθ as $\frac{\sin\theta}{\cos\theta}$sinθcosθ, which gives us the following graph:

$y=\tan\theta$y=tanθ

 

Worked example

example 1

By using the graph of $y=\cos x$y=cosx, what is the sign of $\cos345^\circ$cos345°?

Think: Using the graph of $y=\cos x$y=cosx, we can roughly estimate where the point $\left(345^\circ,\cos345^\circ\right)$(345°,cos345°) lies and from this, determine the sign of $\cos345^\circ$cos345°.

Do: We plot the point on the graph of $y=\cos x$y=cosx below.

The point $\left(345^\circ,\cos345^\circ\right)$(345°,cos345°) drawn on the graph of $y=\cos x$y=cosx.

 

We can observe that the height of the curve at this point is above the $x$x-axis, and that $\cos345^\circ$cos345° is positive.

example 2

What quadrant does an angle with measure $345^\circ$345° lie in?

Think: $345^\circ$345° lies between $270^\circ$270° and $360^\circ$360°.

Do: An angle with a measure that lies between $270^\circ$270° and $360^\circ$360° is said to be in the fourth quadrant. So angle with measure $345^\circ$345° lies in quadrant $IV$IV.

Reflect: The value of $\cos$cos is positive in the first and fourth quadrant and negative in the second and third quadrant. This holds true when we look at the graph of $y=\cos x$y=cosx as well.

Practice questions

question 1

Consider the equation $y=\sin x$y=sinx.

  1. Using the fact that $\sin30^\circ=\frac{1}{2}$sin30°=12, what is the value of $\sin150^\circ$sin150°?

  2. Using the fact that $\sin30^\circ=\frac{1}{2}$sin30°=12, what is the value of $\sin210^\circ$sin210°?

  3. Using the fact that $\sin30^\circ=\frac{1}{2}$sin30°=12, what is the value of $\sin330^\circ$sin330°?

  4. Complete the table of values giving answers in exact form.

    $x$x $0^\circ$0° $30^\circ$30° $90^\circ$90° $150^\circ$150° $180^\circ$180° $210^\circ$210° $270^\circ$270° $330^\circ$330° $360^\circ$360°
    $\sin x$sinx $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  5. Plot the graph of $y=\sin x$y=sinx.

    Loading Graph...

question 2

Consider the equation $y=\cos x$y=cosx.

  1. Complete the table of values, giving answers in exact form.

    $x$x $0^\circ$0° $60^\circ$60° $90^\circ$90° $120^\circ$120° $180^\circ$180° $240^\circ$240° $270^\circ$270° $300^\circ$300° $360^\circ$360°
    $\cos x$cosx $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  2. Plot the graph of $y=\cos x$y=cosx.

    Loading Graph...

question 3

Given the unit circle, which two of the following is true about the graph of $y=\tan x$y=tanx?

  1. The graph of $y=\tan x$y=tanx repeats in regular intervals since the values of $\sin x$sinx and $\cos x$cosx repeat in regular intervals.

    A

    The graph of $y=\tan x$y=tanx is defined for any measure of $x$x.

    B

    Since the radius of the circle is one unit, the value of $y=\tan x$y=tanx lies in the region $-1\le y\le1$1y1.

    C

    The range of values of $y=\tan x$y=tanx is $-\infty<y<.

    D

Outcomes

11U.D.2.3

Make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the relationship between angles from 0º to 360º and the corresponding sine ratios or cosine ratios, with or without technology, defining this relationship as the function f(x) =sinx or f(x) =cosx, and explaining why the relationship is a function

11U.D.2.4

Sketch the graphs of f(x) =sinx and f(x) =cosx for angle measures expressed in degrees, and determine and describe their key properties

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