Math108 – Handout 6.5
y = sin x
y = cos x
1
-10
1
-5
5
10
-10
-5
5
-1
-1
y = arcsin x iff. sin y = x
y = arccos x iff. cos y = x
1  x  1


2
 y 
10
1  x  1

0  y  
2
2
3
1
2
-2
2
1
-1
-2
2
-2
-1
y = arcsin x
y = arccos x
y = arctan x iff. tan y = x
y = tan x
y = arctan x
x 
6
4

2
-5
5

2
 y 
2

2
1
-2
2
-2
-1
-4
-2
-6
Inverse Properties:
If  1  x  1 and
If  1  x  1
If
x 




2
, then sin(arcsin x) = x and arcsin(sin y) = y.
0  y   , then cos(arccos
and
and
2
 y 

2
 y 

2
x) = x and arccos(cos y) = y.
, then tan(arctan x) = x and arctan(tan y) = y.
Evaluate each of the following EXACTLY (No Decimals). Give any angles in radians.
Remember:
y = arcsin x = sin1 x
1  x  1


2
 y 

2
y = arccos x = cos1 x
1  x  1
0  y  
y = arctan x = tan1 x
x  



 y 
2
2
 1
   = __________
 2
(a) arccos
2
= ____________
2
(b) arccos 0 = ____________
(c) arccos
(d) arctan
3
= ____________
3
(e) arccos 1 = _____________
(f) arcsin  

(g) arctan 
(j) arcsin

3 = __________
3
= ____________
2
(h) arcsin
(i) arctan 1 = _____________



(l) arctan 0 = _____________
(k) arcsin  
3
 = _____________
2 

) = ____________
3
(n) cos(arccos (.17)) = ____________
(p) arctan (tan

) = ____________
9
(q) arccos(cos
(s) tan(arctan
7
) = ____________
10
3
) = ____________
2
(y) arccos( 3 sin

) = ____________
6
2
 = __________
2 
 1
   = __________
 2
(m) arcsin(sin
(v) cos(arcsin




) = ____________
4
(t) arcsin(sin 1) = ____________
(o) tan(arctan .25) = ____________
(r) sin(arcsin (.18)) = ____________
(u) cos(arccos
3
) = ____________
5

) = ____________
2
(w) arccos(tan

) = ____________
2
(x) arctan(sin
(z) arctan(2sin

) = ____________
3
() arctan(tan
11
) = ____________
6
Build right triangles for each of the following and use the triangles to find the following trig function values below. Then,
label the lengths of all sides of the triangle accordingly. Be sure to label the angle inside the triangle for the corresponding
information.
(a) arctan
5
12
(a) sin(arctan
(c) arccos
(b) arcsin
5
) = __________________
12
6
x
(c) cot(arccos
3
5
(b) cos(arcsin
3
) = ___________________
5
(d) arctan 3x
6
) = ___________________
x
(d) cos(arctan 3x) = __________________
Evaluate each of the following EXACTLY (No Decimals).
11
) = ____________
13
(b) sec(arccos
7
) = ___________
9
(c) cot(arctan
15
) = ____________
2
(d) cot(arctan 5) = ____________
(e) csc(arcsin
1
) = ____________
6
(f) sec(arccos
5
) = ____________
13
(g) sec(arccos 5) = ____________
(h) csc(arcsin
9
) = ____________
8
(i) cot(arctan
15
) = ____________
14
(k) sec(arctan
2
) = ____________
x
(l) sin(arccos
x
) = ____________
4
(a) csc(arcsin
(j) tan (arcsin
3
) = ____________
5