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.
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.
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| 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°≤θ≤90°. 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.
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| Definition of $\cos$cos and $\sin$sin can extend beyond $0^\circ\le\theta\le90^\circ$0°≤θ≤90°. |
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:
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| $y=\cos\theta$y=cosθ |
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| $y=\sin\theta$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:
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| $y=\tan\theta$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.
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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.
Using the fact that $\sin30^\circ=\frac{1}{2}$sin30°=12, what is the value of $\sin150^\circ$sin150°?
Using the fact that $\sin30^\circ=\frac{1}{2}$sin30°=12, what is the value of $\sin210^\circ$sin210°?
Using the fact that $\sin30^\circ=\frac{1}{2}$sin30°=12, what is the value of $\sin330^\circ$sin330°?
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 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Plot the graph of $y=\sin x$y=sinx.
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question 2
Consider the equation $y=\cos x$y=cosx.
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 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Plot the graph of $y=\cos x$y=cosx.
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question 3
Given the unit circle, which two of the following is true about the graph of $y=\tan x$y=tanx?
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.
AThe graph of $y=\tan x$y=tanx is defined for any measure of $x$x.
BSince the radius of the circle is one unit, the value of $y=\tan x$y=tanx lies in the region $-1\le y\le1$−1≤y≤1.
CThe range of values of $y=\tan x$y=tanx is $-\infty
D