SME
Grade 11 Math · Problem type
A point on the terminal side
Use this for “(−3, 4) is on the terminal side of θ. Find sin θ, cos θ and tan θ.”
Diagram
r
=
√(
(−3)
2
+
4
2
)
=
√
25
=
5
cos θ = x/r = −3/5
sin θ = y/r = 4/5
tan θ = y/x = −4/3
The unit point (cos θ, sin θ) is (x, y) scaled by 1/r: θ ≈ 126.87°
This is an example. Type your own values to start a new problem.
x
x of the point
example
y
y of the point
example
r
Distance from the origin
worked out (not typed)
sin θ
sin θ
worked out (not typed)
cos θ
cos θ
worked out (not typed)
tan θ
tan θ
worked out (not typed)
Formulas
r
=
√
x
2
+
y
2
5
=
√(
(−3)
2
+
4
2
)
sin
θ
=
y
r
4
5
=
4
5
cos
θ
=
x
r
−
3
5
=
−
3
5
tan
θ
=
y
x
−
4
3
=
4
−3
Assumptions
•
The point (x, y) is r = √(x² + y²) from the origin, on the ray that ends θ.
•
Scaled by 1/r it lands on the unit circle, at (x/r, y/r) = (cos θ, sin θ).
•
So sin θ = y/r, cos θ = x/r and tan θ = y/x; the signs of x and y set the quadrant.
Step-by-step
WE KNOW
x = −3, y = 4
FIND
distance from the origin (r), sin θ (sin θ), cos θ (cos θ), tan θ (tan θ)
Step 1 · Find distance from the origin (r)
Use r = √(x² + y²)
The distance from the origin, by the Pythagorean theorem.
r
=
√(
(−3)
2
+
4
2
)
r
=
√
9 + 16
r
=
√
25
r
=
5
Step 2 · Find sin θ (sin θ)
Use sin θ = y ÷ r
The
y
of
the
unit
point:
the
point
scaled
by
1
r
.
sin
θ
=
4
5
sin
θ
=
4
5
Step 3 · Find cos θ (cos θ)
Use cos θ = x ÷ r
The
x
of
the
unit
point.
cos
θ
=
−
3
5
cos
θ
=
−
3
5
Step 4 · Find tan θ (tan θ)
Use tan θ = y ÷ x
Rise
over
run
along
the
ray;
with
x
=
0
the
ray
is
upright
and
tan
θ
has
no
value.
tan
θ
=
4
−3
tan
θ
=
−
4
3
Check
Put the numbers back in. Both sides should match.
5
=
√(
(−3)
2
+
4
2
)
✓
4
5
=
4
5
✓
−
3
5
=
−
3
5
✓
−
4
3
=
4
−3
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Refresh
Arc length, sector area and radian measure
The Pythagorean theorem's converse and special right triangles