1. Unit Circle – Learning Outcomes
Define sin 𝜃 and cos 𝜃 for all values of 𝜃.
Define tan 𝜃 .
Solve problems about the unit circle within the four quadrants.
Solve problems about the unit circle for 𝜃< 0 𝑜 and 𝜃> 360 𝑜 .

2. Define sin 𝜃 , cos 𝜃 , and tan 𝜃
T&T4 – pg 41-44
Define sin 𝜃 , cos 𝜃 , and tan 𝜃
The right-angled triangle definitions of sine, cosine, and tangent limit the possible angles between 0 and 90 degrees.
To generalise the trig functions, we redefine them based on the unit circle.
The unit circle is a circle on the coordinate plane with centre 0, 0 and radius 1.

3. Define sin 𝜃 , cos 𝜃 , and tan 𝜃
T&T4 – pg 41-44
Define sin 𝜃 , cos 𝜃 , and tan 𝜃
Measuring the angle anti-clockwise from the positive 𝑥-axis:
cos 𝐴 is the 𝑥-coordinate where the radius meets the circumference,
sin 𝐴 is the 𝑦-coordinate where the radius meets the circumference.
tan 𝐴 = sin 𝐴 cos 𝐴

4. Solve Quadrant Problems
T&T4 – pg 41-44
Solve Quadrant Problems
Using the unit circle, complete this table.
𝑨 (degrees) 90
180
270
360
sin 𝐴
cos 𝐴
tan 𝐴
Recall that angles measure anti-clockwise from the positive 𝑥-axis.
cos 𝐴 is the 𝑥-coordinate
sin 𝐴 is the 𝑦-coordinate
tan 𝐴 = sin 𝐴 cos 𝐴

5. Solve Quadrant Problems
T&T4 – pg 41-44
Solve Quadrant Problems
Draw a unit circle including axes.
Draw radiuses at 60o, 120o, 240o, and 300o.
Find cos 𝐴 for each of the angles drawn.
Find sin 𝐴 for each of the angles drawn.

6. Solve Quadrant Problems
T&T4 – pg 41-44
Solve Quadrant Problems
Each angle can be reframed as the smallest angle made with the 𝑥- axis, called the reference angle.
For 60 𝑜 , 120 𝑜 , 240 𝑜 , and 𝑜 , this angle is 60 𝑜 .
Since cos 60 =0.5, cosine of each of the other angles is either 0.5 or −0.5 depending on what side of the circle it’s on.

7. Solve Quadrant Problems
T&T4 – pg 41-44
Solve Quadrant Problems
Recall that cos 𝐴 represents the 𝑥- coordinate of the circumference.
Thus, it is positive in the first and fourth quadrants of the circle.
Similarly, it is negative in the second and third quadrants of the circle.

8. Solve Quadrant Problems
T&T4 – pg 41-44
Solve Quadrant Problems
Similarly, sin 60 =0.866, so the sine of each of the other angles is either or −0.866.

9. Solve Quadrant Problems
T&T4 – pg 41-44
Solve Quadrant Problems
Recall that sin 𝐴 represents the 𝑦- coordinate of the circumference.
Thus, it is positive in the first and second quadrants of the circle.
Similarly, it is negative in the third and fourth quadrants of the circle.

10. Solve Quadrant Problems
T&T4 – pg 41-44
Solve Quadrant Problems
Recall that tan 𝐴 represents sin 𝐴 cos 𝐴 .
Thus, it is positive in the first and third quadrants of the circle
Similarly, it is negative in the second and fourth quadrants of the circle.
(You can think of it as the slope of the radius).

11. Solve Quadrant Problems
T&T4 – pg 41-44
Solve Quadrant Problems
If cos 𝐴 = , find two values of 𝐴 if 0 𝑜 ≤𝐴≤ 360 𝑜 .
If sin 𝐵 =− 1 2 , find two values of 𝐵 if 0 𝑜 ≤𝐵≤ 360 𝑜 .
If tan 𝐶 = 3 , find two values of 𝐶 if 0 𝑜 ≤𝐶≤ 360 𝑜 .

12. Solve Quadrant Problems
T&T4 – pg 41-44
Solve Quadrant Problems
2006 HL P2 Q4
Find the value of 𝜃 for which cos 𝜃 =− , 0 𝑜 ≤𝜃≤ 180 𝑜
2006 HL P2 Q4
Write down the values of 𝐴 for which
cos 𝐴 = 1 2 , where 0 𝑜 ≤𝐴≤ 360 𝑜
1998 HL P2 Q4
Find the values of 𝜃 for which
cos 𝜃 = , where 0 𝑜 ≤𝜃≤ 360 𝑜
2011 HL P2 Q5
Find the values of 𝑥 for which
3 tan 𝑥 = 3 , where 0 𝑜 ≤𝑥≤ 360 𝑜

13. Solve > 360 𝑜 and < 0 𝑜 Problems
T&T4 – pg 67-70
Solve > 360 𝑜 and < 0 𝑜 Problems
Sine, cosine, and tangent are periodic.
Each function repeats itself every 360 𝑜 .
e.g. cos 400 = cos (400−360) = cos 140 .
In general,
cos 𝐴 = cos 𝐴+360𝑛
sin 𝐴 = sin (𝐴+360𝑛)
tan 𝐴 =tan⁡(𝐴+360𝑛)

14. Solve > 360 𝑜 and < 0 𝑜 Problems
T&T4 – pg 67-70
Solve > 360 𝑜 and < 0 𝑜 Problems
Likewise, if we measure negative angles (going clockwise instead of anti- clockwise), the result is equivalent to some anti- clockwise angle.
In general,
cos 𝐴 = cos 𝐴+360𝑛
sin 𝐴 = sin (𝐴+360𝑛)
tan 𝐴 =tan⁡(𝐴+360𝑛)
with negative 𝑛

15. Solve > 360 𝑜 and < 0 𝑜 Problems
T&T4 – pg 67-70
Solve > 360 𝑜 and < 0 𝑜 Problems
Find:
tan 495 𝑜
sin 840 𝑜
cos (− 120 𝑜 )

16. Solve > 360 𝑜 and < 0 𝑜 Problems
T&T4 – pg 67-70
Solve > 360 𝑜 and < 0 𝑜 Problems
e.g. If cos 3𝐴 =0.5, find all values of 𝐴 if 0 𝑜 ≤𝐴≤ 360 𝑜 .
cos − = 60 𝑜 , so our reference angle is 60 𝑜 .
Cosine is positive in the first and fourth quadrants, so: 𝑛
𝟑𝑨=𝟔𝟎+𝟑𝟔𝟎𝒏
𝟑𝑨=𝟑𝟎𝟎+𝟑𝟔𝟎𝒏
3𝐴=60
⇒𝐴= 20 𝑜
3𝐴=300
⇒𝐴= 100 𝑜 1
3𝐴=60+360
⇒3𝐴=420
⇒𝐴= 140 𝑜
3𝐴=
⇒3𝐴=660
⇒𝐴= 220 𝑜 2
3𝐴=60+720
⇒3𝐴=780
⇒𝐴= 260 𝑜
3𝐴=
⇒3𝐴=1020
⇒𝐴= 340 𝑜

17. Solve > 360 𝑜 and < 0 𝑜 Problems
T&T4 – pg 67-70
Solve > 360 𝑜 and < 0 𝑜 Problems
If cos 2𝐴 = , find two values of 𝐴 if 0 𝑜 ≤𝐴≤ 360 𝑜 .
If sin 3𝐵 =− 1 2 , find two values of 𝐵 if 0 𝑜 ≤𝐵≤ 360 𝑜 .
If tan 4𝐶 = 3 , find two values of 𝐶 if 0 𝑜 ≤𝐶≤ 360 𝑜 .