7.5
RIGHT TRIANGLES:
INVERSE TRIGONOMETRIC
FUNCTIONS
Functions Modeling Change:
A Preparation for Calculus,
4th Edition, 2011, Connally
The Inverse Sine Function
For 0 ≤ x ≤ 1:
arcsin x = sin−1 x =
The angle in a right triangle whose sine is x.
Example 2
Use the inverse sine function to find
the angles in the figure.
φ
5
3
θ
4
Solution
Using our calculator’s inverse sine function:
sin θ = 3/5 = 0.6 so θ = sin−1(0.6) = 36.87◦
sin φ = 4/5 = 0.8 so φ = sin−1(0.8) = 53.13◦
Functions Modeling Change:
A Preparation for Calculus,
4th Edition, 2011, Connally
The Inverse Tangent Function
arctan x = tan−1 x = The angle in a right triangle whose tan is x.
Example 3
The grade of a road is 5.8%. What angle does the road make with
the horizontal?
Solution
Since the grade is 5.8%, the road climbs 5.8 feet for 100 feet; see
the figure. We see that
tan θ = 5.8/100 = 0.058.
So
θ = tan−1(0.058) = 3.319◦
5.8 ft
using a calculator.
θ
100 ft
A road rising at a grade of 5.8% (not to scale)
Functions Modeling Change:
A Preparation for Calculus,
4th Edition, 2011, Connally
Summary of
Inverse Trigonometric Functions
We define:
• the arc sine or inverse sine function as
arcsin x = sin−1 x = The angle in a right triangle whose sine is x
• the arc cosine or inverse cosine function as
arccos x = cos−1 x = The angle in a right triangle whose cosine is x
• the arc tangent or inverse tangent function as
arctan x = tan−1 x = The angle in a right triangle whose tangent is x.
This means that for an angle θ in a right triangle (other than the
right angle),
sin θ = x means θ = sin−1 x
cos θ = x means θ = cos−1 x
tan θ = x means θ = tan−1 x.
Functions Modeling Change:
A Preparation for Calculus,
4th Edition, 2011, Connally