Precalculus, Section 7.5, #78
Sum and Difference Formulas
Find the exact value of the expression.1
3
5
− sin−1 −
cos tan−1
12
5
The instructions ask for the exact value, so this is not a calculator exercise.
5
5
.
is the angle between − π2 and π2 whose tangent is 12
Recall that tan−1 12
We draw a triangle with an angle θ such that
5
tan (θ) = 12
. Using the Pythagorean theorem to find
the hypotenuse gives us the diagram at right.
13
5
θ
12
Similarly, sin−1 − 35 is the angle between − π2 and
π
2
whose sine is − 53 .
We draw a triangle with an angle φ such that
sin (φ) = − 53 . Using the Pythagorean theorem to find
the adjacent leg gives us the diagram at right.
4
φ
5
-3
Now that we have representations of the inverse trig functions, we can evaluate the given expression. Since
this expression includes cos (α − β), we’ll probably need to recall
cos (α − β) = cos (α) cos (β) + sin (α) sin (β)
So
3
3
5
5
−1
−1
−1
−1
−
−
cos tan
− sin
= cos tan
· cos sin
12
5
12
5
3
5
· sin sin−1 −
+ sin tan−1
12
5
and from the triangles above,
12
13
48
=
65
33
=
65
=
1 Sullivan,
4
5
3
+
·−
5 13
5
15
−
65
·
Precalculus: Enhanced with Graphing Utilities, p. 486, #78.