1. Bell Ringer
Yola just planted a Hybrid Elm tree. The nursery claims the tree grows 12 feet per year. Yola wants to verify the claim. She walks 100 feet from the base of the tree and, using a transit that is 2 feet off the ground, determines the angle of elevation to the top of the tree is 5.7o. One year later, the angle of elevation 100 feet from the tree is 11.9o. Is the nursery’s claim true?

2. h = height of tree above transit g = growth of tree during first year
100 feet
2 feet
h = height of tree above transit
g = growth of tree during first year

3. Height of tree after 1 year: 2 + 9.98 + 11.09 = 23.07 feet

4. 4.4: Trig of Any Angle Objectives:
To find trig values of an angle given any point on the terminal side of an angle
To find the acute reference angle of any angle

5. Unit Circle Rationale
Recall that when using the unit circle to evaluate the value of a trig function, cos θ = x and sin θ = y. What we didn’t point out is that since the radius (hypotenuse) is 1, the trig values are really cos θ = x/1 and sin θ = y/1. So what if the radius (hypotenuse) is not 1?

6. Exercise 1
Let θ be an angle whose terminal side contains the point (4, 3). Find sin θ, cos θ, and tan θ.

7. Exercise 1
Let θ be an angle whose terminal side contains the point (4, 3). Find sin θ, cos θ, and tan θ.
Realize that the triangle formed by the point (4, 3) is similar to a triangle in the unit circle. To get to that unit circle triangle, we would have to scale down the larger triangle by dividing by the scale factor. In this case, that’s 5, the length of the larger hypotenuse.

8. Trig of Any Angle
Let (x, y) be a point on the terminal side of an angle θ in standard position with

9. Exercise 2
The previous definitions imply that tan θ and sec θ are not defined when x = 0. So what values of θ are we talking about? They also imply that cot θ and csc θ are not defined when y = 0. So what values of θ are we talking about?

10. Exercise 3
Let θ be an angle whose terminal side contains the point (−2, 5). Find the six trig functions for θ.

11. Signs of Trig Functions
We can find the sign of a particular trig function based on which quadrant (x, y) lies within.

12. Exercise 4
Given sin θ = 4/5 and tan θ < 0, find cos θ and csc θ.

13. Reference Angles
When building the unit circle, for 120° we drew a triangle with the x-axis to form a 60° angle. This 60° angle was the reference angle for 120°.

14. Reference Angles
If we connect the trig of any angle to right triangle trigonometry, we need a reference angle that is acute to be able to evaluate the function.

15. Reference Angles
Let θ be an angle in standard position. It’s reference angle is the acute angle θ’ formed by the terminal side of θ and the x-axis.

16. Reference Angles
Let θ be an angle in standard position. It’s reference angle is the acute angle θ’ formed by the terminal side of θ and the x-axis.

17. Exercise 5

18. Reference Angles
When your angle is negative or is greater than one revolution, to find the reference angle, first find the positive coterminal angle between 0° and 360° or 0 and 2π.

19. Reference Angles How Reference Angles Work:
Same except maybe a difference of sign.

20. Reference Angles To find the value of a trig function of any angle:
Find the trig value for the associated reference angle
Pick the correct sign depending on where the terminal side lies

21. Exercise 6
Evaluate:
sin 5π/3
cos (−60°)
tan 11π/6

22. Exercise 7
Let θ be an angle in Quadrant III such that sin θ = −5/13. Find a) sec θ and b) tan θ using trig identities.