UK Secondary (7-11)
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Intro to sin(x), cos(x) and tan(x)
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 anticlockwise 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 of the following is true about the graph of $y=\tan x$y=tanx?

Select all that apply.

  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

    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

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