SME
Grade 12 Math · Problem type
The cosine of an inverse sine
Use this for “Find the exact value of cos(sin⁻¹(3/5)).”
Diagram
θ = 36.87°: the point (cos θ, sin θ) is ≈ (0.8, 0.6).
This is an example. Type your own values to start a new problem.
cos(sin⁻¹(
x
))
=
1
−
(
)
2
=
4
5
y
A
The angle sin⁻¹(x)
worked out (not typed) · °
Formulas
A
=
sin⁻¹(
x
)
36.8699
=
sin⁻¹(
3
5
)
y
=
√
1 −
x
2
4
5
=
√(1
−
(
3
5
)
2
)
Assumptions
•
Let A = sin⁻¹(x): an angle from −90° to 90° with sin A = x.
•
Then cos A = √(1 − x²), from sin²A + cos²A = 1.
•
The root is positive because A is from −90° to 90°, where the cosine is not negative.
Step-by-step
WE KNOW
x = 3/5
FIND
the angle sin⁻¹(x) (A), cos(sin⁻¹(x)) (y)
Step 1 · Find the angle sin⁻¹(x): A
Use A = sin⁻¹(x)
The
one
angle
from
−90°
to
90°
whose
sine
is
x
.
A
=
sin⁻¹(
3
5
)
A
=
36.8699°
Step 2 · Find cos(sin⁻¹(x)): y
Use y = √(1 − x²)
cos
A
=
√
1 − sin²
A
,
and
sin
A
=
x
;
the
root
is
positive
since
A
is
from
−90°
to
90°.
y
=
√(1
−
(
3
5
)
2
)
y
=
√
1 − 0.36
y
=
√
0.64
y
=
4
5
Check
Put the numbers back in. Both sides should match.
sin(36.8699°)
=
3
5
✓
4
5
=
√(1
−
(
3
5
)
2
)
✓
← Previous lesson
Inverse tangent: the angle of a slope
Next lesson →
Sum, difference and double-angle formulas; trigonometric equations
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.