Trig equations & identities

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The Unit Circle

Sine and Cosine on the Unit Circle

The key idea with triangles who's hypotenuse lie on the unit circle (that form an angle of θ with the x-axis) is that cosθ represents length of the base, and sinθ represents the height.

Take a look at the graph below and notice the following relationships always hold:

cosθsinθ=x-coordinate=y-coordinate

Checkpoint

Find the coordinates of P.

Select the correct option

Key values of Sin, Cos & Tan

The following table shows the values of sinθ and cosθ for the so called critical angles θ. These are angles that give "nice" values for sin and cos:

θ (rad)	sinθ	cosθ
0	0	1
6π	21	2√3
4π	2√2	2√2
3π	2√3	21
2π	1	0

It might be helpful when memorizing these to notice:

θ sinθ cosθ0 √40 √446π √41 √434π √42 √423π √43 √412π √44 √40

Checkpoint

Find

cos(6π)
sin(3π)

Select the correct option

Before we continue, it's useful to introduce the terminology of quadrants. These are simply the four sections that the coordinate axes divide the plane into:

Discussion

The diagram below shows a point P in quadrant II:

Is its x-value positive or negative? What does that say about cosθ?
Is its y-value positive or negative? What does that say about sinθ?
What can you deduce about sin and cos in the other quadrants.; I,III and IV?

Part (a)

We see that P lies to the left of the y-axis, so cosθ<0.

Part (b)

We see that P lies above the y-axis, so sinθ>0, since sinθ

Part (c)

In each quadrant the sign of x (so of cosθ) and of y (so of sinθ) is:

I:III:IV:x>0,y>0⟹cosθ>0,sinθ>0x<0,y<0⟹cosθ<0,sinθ<0x>0,y<0⟹cosθ>0,sinθ<0

Quadrants

The unit circle can be divided into quadrants based on the sign of cosθ and sinθ. These correspond to the 4 quadrants produced by the intersection of the x and y axes. The quadrants are denoted Q1, Q2, Q3 and Q4.

Quadrant	sin	cos
Q1	+	+
Q2	+	-
Q3	-	-
Q4	-	+

Checkpoint

Determine the sign (positive/negative) of each of the following:

sin(105°)
cos(713π)

Select the correct option

Discussion

You're standing on the unit circle, and then make one full lap to return to the same spot.

How many radians have you rotated?
What might this suggest about sin(θ) and sin(θ+2π)?
What might this suggest about cos(θ) and cos(θ+2π)?

Part (a)

The angle θ (in radians) corresponding to an arc of length s on a circle of radius r satisfies

s=rθ

On the unit circle (r=1), one full lap has arc length equal to the circumference,

s=2πr=2π⋅1=2π

Hence

θ=rs=12π=2π

Part (b)

On the unit circle a point at angle θ is (cosθ,sinθ).

After adding a full turn of 2π the point at angle θ+2π is the same, so

(cos(θ+2π),sin(θ+2π))=(cosθ,sinθ)

Equating the second coordinates gives

sin(θ+2π)=sinθ

Thus sin is periodic with period 2π.

Part (c)

On the unit circle a point at angle θ is (cosθ,sinθ).

After adding a full turn of 2π the point at angle θ+2π is the same, so

(cos(θ+2π),sin(θ+2π))=(cosθ,sinθ)

Equating the first coordinates gives

cos(θ+2π)=cosθ

Thus cos is periodic with period 2π.

Periodicity

Since a full circle is 2π radians, adding 2π to any angle θ gives the same point on the unit circle. In fact, adding any integer multiple of 2π gives the same point:

cos(θ+2kπ)sin(θ+2kπ)=cosθ=sinθk∈Z🚫

Checkpoint

Find the value of cos(367π).

Select the correct option

Symmetry About the X-axis

sin(−θ)cos(−θ)=−sinθ=cosθ🚫

Checkpoint

If θ=−6π, find sinθ and cosθ.

Select the correct option

Discussion

Using the animation above, can you derive simplified expressions for sin(π−θ) and cos(π−θ)?

From the diagram, the point P has coordinates (cosθ,sinθ). Its mirror image Q across the y–axis has coordinates

(−cosθ,sinθ).

On the other hand, Q also lies on the unit circle at angle π−θ, so

Q=(cos(π−θ),sin(π−θ)).

Equating components gives

cos(π−θ)sin(π−θ)=−cosθ=sinθ

Symmetry About the Y-axis

sin(π−θ)cos(π−θ)=sinθ=−cosθ🚫

Checkpoint

If θ=65π, find sinθ and cosθ.

Select the correct option

Discussion

Using the animation above, can you derive simplified expressions for sin(θ+π) and cos(θ+π)?

A point on the unit circle at angle θ has coordinates (cosθ,sinθ). Adding π takes you to the diametrically opposite point (−cosθ,−sinθ), so

cos(θ+π)=−cosθ,sin(θ+π)=−sinθ

Equivalently, by the addition formulas and using cosπ=−1,sinπ=0:

sin(θ+π)cos(θ+π)=sinθcosπ+cosθsinπ=sinθ⋅(−1)+cosθ⋅0=−sinθ=cosθcosπ−sinθsinπ=cosθ⋅(−1)−sinθ⋅0=−cosθ

Symmetry About the Origin

sin(θ+π)cos(θ+π)=−sinθ=−cosθ🚫

Checkpoint

If θ=45π, find sinθ and cosθ.

Select the correct option

Relating Angles Between Quadrants

Once the values of sin and cos are memorized for critical angles in the first quadrant, you can use the rules of symmetry and periodicity to determine the values of critical angles in arbitrary quadrants.

Examples

cos(631π) =cos(5π+6π)=cos(π+6π)=−cos(6π)=−2√3

sin(4223π) =sin(4220π+3π)=sin(55π+43π)=sin(π+43π)=−sin(43π)=−2√2

Exercise

Find

cos(3113π)
sin(−459π)

Select the correct option

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