Intro to sin(x), cos(x) and tan(x)
The definitions of the trigonometric functions $$sin, $$cos and $$tan that we've seen use the ratios of side lengths of a right-angled triangle. More specifically, we call this the right-angled triangle definition of the trigonometric functions, but there are other methods to define these functions more broadly.
For a right-angled triangle, where $$θ is the measure for one of the angles (excluding the right angle), we have that:
| $$sinθ | $$= | $$opposite hypotenuse |
| $$cosθ | $$= | $$adjacent hypotenuse |
| $$tanθ | $$= | $$opposite adjacent |
Now consider a right-angled triangle, with hypotenuse that has a length of one unit with a vertex centred at the origin. We can construct a unit circle around the triangle as shown below.
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| A right-angled triangle inscribed in the unit circle. |
The point indicated on the circle has coordinates $$(cosθ,sinθ) using the right-angled triangle definition of $$cos and $$sin. Unfortunately, this definition is limited to angles with measures in the range of $$0°≤θ≤90°. However, more broadly, we can use the unit circle to define $$cos and $$sin for angles with any measure. We call this the unit circle definition. In this definition, the value of these functions will be the $$x- and $$y-values of a point on the unit circle after having rotated by an angle of measure $$θ in the anticlockwise direction as shown below. If $$θ is negative then the point is rotated in the clockwise direction.
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| Definition of $$cos and $$sin can extend beyond $$0°≤θ≤90°. |
As we move through different values of $$θ the value of $$cosθ and $$sinθ move accordingly between $$−1 and $$1. If we plot the values of $$cosθ and $$sinθ according to different values of $$θ on the unit circle, we get the following graphs:
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| $$y=cosθ |
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| $$y=sinθ |
As in the right-angled triangle definition, we still define $$tanθ as $$sinθcosθ, which gives us the following graph:
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| $$y=tanθ |
Worked example
example 1
By using the graph of $$y=cosx, what is the sign of $$cos345°?
Think: Using the graph of $$y=cosx, we can roughly estimate where the point $$(345°,cos345°) lies and from this, determine the sign of $$cos345°.
Do: We plot the point on the graph of $$y=cosx below.
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The point $$(345°,cos345°) drawn on the graph of $$y=cosx. |
We can observe that the height of the curve at this point is above the $$x-axis, and that $$cos345° is positive.
example 2
What quadrant does an angle with measure $$345° lie in?
Think: $$345° lies between $$270° and $$360°.
Do: An angle with a measure that lies between $$270° and $$360° is said to be in the fourth quadrant. So angle with measure $$345° lies in quadrant $$IV.
Reflect: The value of $$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 $$y=cosx as well.
Practice questions
question 1
Consider the equation $$y=sinx.
Using the fact that $$sin30°=12, what is the value of $$sin150°?
Using the fact that $$sin30°=12, what is the value of $$sin210°?
Using the fact that $$sin30°=12, what is the value of $$sin330°?
Complete the table of values giving answers in exact form.
$$x $$0° $$30° $$90° $$150° $$180° $$210° $$270° $$330° $$360° $$sinx $$ $$ $$ $$ $$ $$ $$ $$ $$ Plot the graph of $$y=sinx.
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question 2
Consider the equation $$y=cosx.
Complete the table of values, giving answers in exact form.
$$x $$0° $$60° $$90° $$120° $$180° $$240° $$270° $$300° $$360° $$cosx $$ $$ $$ $$ $$ $$ $$ $$ $$ Plot the graph of $$y=cosx.
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question 3
Given the unit circle, which two of the following is true about the graph of $$y=tanx?
The graph of $$y=tanx repeats in regular intervals since the values of $$sinx and $$cosx repeat in regular intervals.
AThe graph of $$y=tanx is defined for any measure of $$x.
BSince the radius of the circle is one unit, the value of $$y=tanx lies in the region $$−1≤y≤1.
CThe range of values of $$y=tanx is $$−∞<y<∞.
D