1. Evaluating Trigonometric Functions for any Angle

2. What You Should Learn Evaluate trigonometric functions of any angle
Evaluate trigonometric functions of real numbers

3. Introduction
Following is the definition of trigonometric functions of Any Angle.

4. Introduction
Because r = cannot be zero, it follows that the sine and cosine functions are defined for any real value of  .
However, when x = 0, the tangent and secant of  are undefined. For example, the tangent of 90  is undefined. Similarly, when y = 0, the cotangent and cosecant of  are undefined.

5. Example 1 – Evaluating Trigonometric Functions
Let (–3, 4) be a point on the terminal side of  (see Figure 4.34). Find the sine, cosine, and tangent of .
Figure 4.34

6. Example 1 – Solution
Referring to Figure 4.34, you can see that x = –3, y = 4, and

7. Example 1 – Solution
So, you have
and
cont’d

8. Introduction
The signs of the trigonometric functions in the four quadrants can be determined easily from the definitions of the functions. For instance, because
it follows that cos  is positive wherever x > 0, which is in Quadrants I and IV.

9. Using the Sign If and lies in Quadrant III, find sin and tan θ θ θ -1
-√3 2

10. Review
Find the exact values of the other five trig functions for an angle θ in standard position, given 12
360o θ -5 13
270o

11. Your Turn:
Let θ be an angle in Quadrant III such that sin θ = −5/13. Find a) sec θ and b) tan θ.

12. Right Triangle Trigonometry
Trigonometry is based upon ratios of the sides of right triangles.
The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle. θ
hyp
opp
adj
The sides of the right triangle are:
 the side opposite the acute angle ,
 the side adjacent to the acute angle ,
 and the hypotenuse of the right triangle.

13. Right Triangle Trigonometry
The hypotenuse is the longest side and is always opposite the right angle.
The opposite and adjacent sides refer to another angle, other than the 90o. A

14. S O H C A H T O A Trigonometric Ratios hyp opp θ adj
The trigonometric functions are:
sine, cosine, tangent, cotangent, secant, and cosecant.
sin Ɵ = cos Ɵ = tan Ɵ =
csc Ɵ = sec Ɵ = cot Ɵ =
opp
hyp
adj
S O H C A H T O A

15. Reciprocal Functions sin  = 1/csc  csc  = 1/sin 
cos  = 1/sec  sec  = 1/cos 
tan  = 1/cot  cot  = 1/tan 

16. Finding the ratios
The simplest form of question is finding the decimal value of the ratio of a given angle.
Find using calculator:
sin 30 =
sin 30 =
cos 23 =
tan 78 =
tan 27 =
sin 68 =

17. Using ratios to find angles
It can also be used in reverse, finding an angle from a ratio.
To do this we use the sin-1, cos-1 and tan-1 function keys.
Example:
sin x = find angle x.
sin-1
0.1115 =
shift
sin ( )
x = sin-1 (0.1115)
x = 6.4o
2. cos x = find angle x
cos-1
0.8988 =
shift
cos ( )
x = cos-1 (0.8988)
x = 26o

18. Calculate the trigonometric functions for  . 4 3 5 
The six trig ratios are
sin  =
sin α =
cos  =
cos α =
tan  =
What is the relationship of
α and θ?
tan α =
cot  =
cot α =
sec  =
They are complementary (α = 90 – θ)
sec α =
csc  =
csc α =

19. Cofunctions sin  = cos (90  ) cos  = sin (90  )
tan  = cot (90  ) cot  = tan (90  )
tan  = cot (π/2  ) cot  = tan (π/2  )
sec  = csc (90  ) csc  = sec (90  )
sec  = csc (π/2  ) csc  = sec (π/2  )

20. Finding an angle from a triangle
To find a missing angle from a right-angled triangle we need to know two of the sides of the triangle.
We can then choose the appropriate ratio, sin, cos or tan and use the calculator to identify the angle from the decimal value of the ratio.
14 cm
6 cm C 1.
Find angle C
Identify/label the names of the sides.
b) Choose the ratio that contains BOTH of the letters.

21. 14 cm
6 cm C 1.
We have been given the adjacent and hypotenuse so we use COSINE:
Cos A = h a
Cos A =
Cos C =
Cos C =
C = cos-1 (0.4286)
C = 64.6o

22. Find angle x 2.
8 cm
3 cm x
Given adj and opp
need to use tan:
Tan A = a o
Tan A =
Tan x =
Tan x =
x = tan-1 (2.6667)
x = 69.4o

23. 3.
12 cm
10 cm y
Given opp and hyp
need to use sin:
Sin A =
sin A =
sin x =
sin x =
x = sin-1 (0.8333)
x = 56.4o

24. Finding a side from a triangle
7 cm k
30o 4.
We have been given the adj and hyp so we use COSINE:
Cos A =
Cos A =
Cos 30 =
Cos 30 x 7 = k
6.1 cm = k

25. 4 cm r
50o 5.
We have been given the opp and adj so we use TAN:
Tan A =
Tan A =
Tan 50 =
Tan 50 x 4 = r
4.8 cm = r

26. 12 cm k
25o 6.
We have been given the opp and hyp so we use SINE:
Sin A =
sin A =
sin 25 =
Sin 25 x 12 = k
5.1 cm = k

27. x = x
5 cm
30o 1.
Cos A =
Cos 30 =
x = cm
4 cm r
50o 2.
Tan 50 x 4 = r
4.8 cm = r
Tan A =
Tan 50 = 3.
12 cm
10 cm y
y = sin-1 (0.8333)
y = 56.4o
sin A =
sin y =
sin y =

28. Example: Given sec  = 4, find the values of the other five trigonometric functions of  .
Solution:
Draw a right triangle with an angle  such that 4 = sec  = = .
adj
hyp θ 4 1
Use the Pythagorean Theorem to solve for the third side of the triangle.
sin  = csc  = =
cos  = sec  = = 4
tan  = = cot  =

29. Applications Involving Right Triangles
A surveyor is standing 115 feet from the base of the Washington Monument. The surveyor measures the angle of elevation to the top of the monument as 78.3. How tall is the Washington Monument?
Solution:
where x = 115 and y is the height of the monument. So, the height of the Washington Monument is
y = x tan 78.3
 115( )  555 feet.

30. Fundamental Trigonometric Identities
Reciprocal Identities
sin  = 1/csc  cos  = 1/sec  tan  = 1/cot  cot  = 1/tan  sec  = 1/cos  csc  = 1/sin 
Co function Identities
sin  = cos(90  ) cos  = sin(90  )
sin  = cos (π/2  ) cos  = sin (π/2  )
tan  = cot(90  ) cot  = tan(90  )
tan  = cot (π/2  ) cot  = tan (π/2  )
sec  = csc(90  ) csc  = sec(90  )
sec  = csc (π/2  ) csc  = sec (π/2  )
Quotient Identities
tan  = sin  /cos  cot  = cos  /sin 
Pythagorean Identities
sin2  + cos2  = 1 tan2  + 1 = sec2  cot2  + 1 = csc2 