The ASTC Circle for Trig

1. The ASTC Circle for Trig

2. Each part of the circle tells you which has a positive answer 90 LL are +ve S A in is +ve 180 0 T C an is +ve os is +ve 270

3. Each part of the circle tells you which has a positive answer. 90 Sin 150 is a positive answer, Cos 150 and Tan 150 both give you negative answers LL are +ve Sin 30, Cos 30 and Tan 30 all give you positive answers A S in is +ve Use your calculator to check the following trig answers (make sure it is in degrees mode 180 0 T C an is +ve os is +ve Tan 200 is a positive answer, Cos 200 and Sin 200 both give you negative answers Cos 300 is a positive answer, Sin 300 and Tan 300 both give you negative answers 270

4. Now if we replaced the angel with a  symbol and gave a positive or negative answer, we could narrow down the possibilities to just two quadrants 90 • Sin  = 0.6 0.6 is a positive answer so it can only be in the quadrants which have Sin having positive answers. Press the mouse to check if my answers match yours • Tan  = -0.2 -0.2 is negative answer so it can only be in the quadrants which tan has negative answers. Press the mouse to check if my answers match yours Sin  = 0.6 S A Sin  = 0.6 Tan  = -0.2 180 0 Tan  = -0.2 T C

5. So how do we solve Trig Equations • We know that we need to get two answers for every equation • We need to make sure that our calculator is set in “degrees mode” • Because we’re finding an angle, we have to be careful to press “Shift” on our calculator.

6. Here’s the trick! • The first step is to use your calculator to work out the answer to the original question. • if Sin  = 0.62 •  = 38.3 (to 1 d.p.) (put this back into your calculator to check 38.3 180 0

7. The second step is to realise that the second answer is in the “S” quadrant • The angle size is either 180–38.3=141.7or 90+38.3= 128.3 • (put this back into your calculator to check!) So the rule is for the “S” quadrant to go 180 -  to get your answer 38.3 180 0

8. It works for Cosine as well! • First step - use your calculator to work out the answer to the original question. • if Cos  = 0.2 •  = 78.5 (to 1 d.p.) (put this back into your calculator to check 78.5 180 0

9. The second step is to realise that the second answer is in the “C” quadrant • The angle size is either • 270 + 78.5 = 348.5 • or • 360 – 78.5 = 281.5 (put this back into your calculator to check!) So the rule is for the “C” quadrant to go 360 -  to get your answer 180 0 78.5

10. I bet I can make it work for Tan! • First step - use your calculator to work out the answer to the original question. • if Tan  = 0.5 •  = 26.6 (to 1 d.p.) (put this back into your calculator to check 26.6 180 0

11. Does the second answer being in the “T” quadrant start to make a pattern • The angle size is either • 180 + 26.6 = 206.6 • or • 270 – 26.6 = 243.4 (put this back into your calculator to check!) So the rule is for the “T” quadrant to go 180 +  to get your answer 180 0 26.6

12. So you should have an idea of the rules now to solve trig equation at the YR 11 Level • You must check for two answers between 0 and 360 degrees • If the question has a positive answer, you should get one answer between 0 and 90 • If the question is Sin with a positive answer, you should get a second answer between 90 and 180 • If the question is Tanwith a positive answer, you should get a second answer between 180 and 270 • If the question is Coswith a positive answer, you should get a second answer between 270 and 360

13. Try These! • Sin  = 0.26 • Cos  = 0.78 • Tan  = 0.07 • Sin  = 0.86 • Cos  = 0.825 • Tan  = 0.36 • Sin  = 0.41 • Cos  = 0.707 • Tan  = 0.717 • Now email me all your answers! Once you have done this I will send you instructions on how to solve trig equations involving negatives (yeah, fun, party!)