E-MAP Mathematics
Section 7.5
Inverse Trig Functions
Functions must be one-to-one in order to have an inverse.
y = sin x
The inverse sine function, written y = sin−1x or y = arcsin x, is
the angle between
and
whose sine is x.
In other words, y = sin−1x if and only if x = sin y and
≤y≤
E-MAP Mathematics
Section 7.5
Find the exact value (in radians) of each of the following:
E-MAP Mathematics
Section 7.5
y = tan x
The inverse tangent function, written y = tan−1x or y = arctan x, is
the angle between
and
whose tangent is x.
In other words, y = tan−1x if and only if x = tan y and
<y<
E-MAP Mathematics
y = cos x
Section 7.5
E-MAP Mathematics
Section 7.5
y = cos x
The inverse cosine function, written y = cos−1x or y = arccos x, is
the angle between 0 and π whose cosine is x.
In other words, y = cos−1x if and only if x = cos y and 0 ≤ y ≤ π
E-MAP Mathematics
Section 7.5
Find the exact value (in radians) of each of the following:
E-MAP Mathematics
Section 7.5
Find the exact value of each of the following:
4
5
−3
E-MAP Mathematics
Section 7.5
Write the expression cos(tan−1(
)) in algebraic form.
1
x
E-MAP Mathematics
Section 7.6
Solve the equation on the interval [0º , 360º)
sin x =
x = 30º
, 150º
E-MAP Mathematics
Section 7.6
Solve the equation on the interval [0 , 2π)
2 cos x − 1 = 0
2 cos x = 1
cos x =
x=
E-MAP Mathematics
Section 7.6
Solving Trig Equations
• Put the equation in terms of one trig function (if
possible).
• Solve for the trig function (using algebra – addition,
subtraction, multiplication, division, factoring).
• Solve for the variable (using inverse trig functions,
reference angles).
E-MAP Mathematics
Section 7.6
Solve the equation on the interval [0º , 360º)
2 cos2x − cos x − 1 = 0
(2 cos x + 1)(cos x − 1) = 0
cos x =
or cos x = 1
x = 120º, 240º, 0º
E-MAP Mathematics
Section 7.6
Find all solutions (in radians) of the following equation
cos θ + sin 2θ = 0
cos θ + 2 sin θ cos θ = 0
cos θ(1 + 2 sin θ) = 0
cos θ = 0 or sin θ =
θ=
θ=
k any integer
E-MAP Mathematics
Section 7.6
Solve the equation on the interval [0º , 360º)
tan 2x = 1
2x = 45º, 225º, 405º, 585º
x = 22.5º, 112.5º, 202.5º, 292.5º