Example #1 and #2 (easy)

1) Use the values secU= -3/2 and tanU > 0 to find the values of all six trig functions.

cos θ  =      1    =    1     =  -2/3 
               sec θ     -3/2  

   
sin²θ  =  1 − cos²θ = 1-(-2/3)² = 1-(4/9) = 5/9

ANSWER
sin u = -√5                                          csc u =   1    =   - 3  
            3                                                       sin u      √5
cos u = -2                                         sec u =  1      =  - 3  
             3                                                    cos u       2
tan u = sin u  =      -√5/3      = √5       cot u = cos u  =   2  
            cos u          -2/3          2                  sin u       √5
 


2) Use the fundamental trig identities to simplify the expression sin x(tan x)

Because of the quotient identity, tan x= sin x
                                                          cosx
sin x(sin x)  = sin²x  
        cos x     cos x

 Example #3 and #4 (medium)

3)Use the fundamental trig identities to simplify the expression   cos²y  
                                                                                         1- sin y
)cos² y= 1 - sin²y   =  (1-sin y)(1+sin y)   =  1+sin y           
            1- sin y           1-sin y

4) Prove that tan x = sec x
                   sin x
First, get everything in terms of sin and cosine
sin x
cos x
    1     
sin x        cos x

sin x   x    1      =    1     
cos x      sin x        cos

Example # 5 and #6 (Challenging)

5) Factor the expression and use the fundamental identities to simplify cos²x sin²x - cos²x

First, factor out a  -cos²x.    -cos²x(-sin²x-1) 
Use the Pythagorean Identity to simplify (-sin²x-1)      -cos²x (-cos²x )= cos^4x²

 
6) Verify that (cscx + cotx) = cotx cscx
                    (tanx + sinx)

First, get everything in terms of sin and cosine.
  + cos
sin    sin  
sin  
+ sin
cos     1


1+ cos   x           cos      
 sin              sin + sin cos

1+cos   x         cos         
  sin             sin (1 +cos)

cos      x       1      =  cot(csc)= cotx cscx
sin               sin


(sin y - 1) ( sin y -1)
Example # 7 and #8 (Real World)
 
7) A ladder is propped against a wall forming an angle x with the ground, so the ladder is the hypotenuse of a right triangle.  If tan x=1 ft what is cos x? How long is the ladder?

Using the Pythagorean Identity, 1 + tan²=sec²:     sec²=1 + (1²)=2     sec=√2
Using the Reciprocal Identity: cos=  1       cos=   1     =   √2  
                                                   sec              √2           2


cos=   adjacent        =       √2               ladder=2 ft                                          
        hypotenuse                2


8) A lean to structure forms a right triangle on the ground.  If csc² is 4/3 ft, how long are the side lenghts of the lean to?
Using the reciprocal identity: sin=  1           sin²=  3      sin=√3         sin=opposite                  side length 1=√3  ft (1.73 ft)        side length 2=2 ft
                                                 csc                  4             2                 hypotenuse                                               

Use the Pythagorean Theorem to find the remaining side:   a²+b²=c²             c²-a²=b²          (2 ft)² - (1.73 ft)² = 1.01 ft   √1.01=1.00 ft 
                                                                                                                                                            side length 3= 1 ft