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
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
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.
1 + 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)
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.
1 + 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
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