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 .

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.

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

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

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.

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 300 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.

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.

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 0.866 or −0.866.

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.

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).

Solve Quadrant Problems T&T4 – pg 41-44 Solve Quadrant Problems If cos 𝐴 = 1 2 , 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 .

Solve Quadrant Problems T&T4 – pg 41-44 Solve Quadrant Problems 2006 HL P2 Q4 Find the value of 𝜃 for which cos 𝜃 =− 3 2 , 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 = 3 2 , 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

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)

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

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 )

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 −1 0.5 = 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=300+360 ⇒3𝐴=660 ⇒𝐴= 220 𝑜 2 3𝐴=60+720 ⇒3𝐴=780 ⇒𝐴= 260 𝑜 3𝐴=300+720 ⇒3𝐴=1020 ⇒𝐴= 340 𝑜

Solve > 360 𝑜 and < 0 𝑜 Problems T&T4 – pg 67-70 Solve > 360 𝑜 and < 0 𝑜 Problems If cos 2𝐴 = 1 2 , 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 .