Question 1

What is sin(45°)?

Accepted Answer

sin(45°) = √2/2 ≈ 0.70710678.

This comes from the 45-45-90 isosceles right triangle, which has two equal legs (both length 1) and a hypotenuse of √2.

sin = opposite / hypotenuse = 1/√2 = √2/2

On the unit circle, 45° is the point (√2/2, √2/2) — exactly halfway up the first quadrant arc. The y-coordinate gives sin(45°) = √2/2.

Note: √2/2 is the same as 1/√2. Both forms are correct; √2/2 is the rationalised form that avoids a square root in the denominator.

Question 2

What is cos(45°)?

Accepted Answer

cos(45°) = √2/2 ≈ 0.70710678.

At 45°, cosine equals sine. This is the only angle in the first quadrant where sin = cos.

cos = adjacent / hypotenuse = 1/√2 = √2/2

The reason sin(45°) = cos(45°) is that 45° is its own complement — 90° - 45° = 45°. The co-function identity says cos(θ) = sin(90° - θ), so cos(45°) = sin(45°).

Geometrically, the point (√2/2, √2/2) lies exactly on the line y = x, which bisects the first quadrant at 45°. The x and y coordinates are identical.

Question 3

What is tan(45°)?

Accepted Answer

tan(45°) = 1.

tan = sin/cos = (√2/2)/(√2/2) = 1

Alternatively: tan = opposite/adjacent = 1/1 = 1 (both legs of the 45-45-90 triangle are equal).

tan(45°) = 1 is a fundamental result. It means the angle makes a slope of 1 — a 45-degree line rises exactly as fast as it runs. This is why 45° is the angle of maximum range for projectile motion (the optimal balance between height and horizontal distance).

The pattern: tan(0°)=0, tan(45°)=1, tan(90°)=undefined. Tan grows from 0 to infinity as the angle goes from 0° to 90°, passing through 1 exactly at 45°.

Question 4

Why does sin(45°) equal cos(45°)?

Accepted Answer

Because 45° is its own complementary angle. The co-function identity states:

sin(θ) = cos(90° - θ)

When θ = 45°: sin(45°) = cos(90° - 45°) = cos(45°)

This only works when θ = 90° - θ, which gives θ = 45°. It is the unique angle in [0°, 90°] where sine and cosine are equal.

From the 45-45-90 triangle perspective: both acute angles are 45°, so the 'opposite' and 'adjacent' sides are the same leg. The numerator in sin and cos is the same length, giving the same ratio to the hypotenuse.

Consequences:
- tan(45°) = sin/cos = 1
- cot(45°) = cos/sin = 1
- csc(45°) = sec(45°) = √2

45° is the most symmetric angle in the first quadrant.

Question 5

What is the 45-45-90 triangle and how is it used?

Accepted Answer

A 45-45-90 triangle is an isosceles right triangle — two equal legs and a hypotenuse. If each leg has length 1, the hypotenuse is √2 (by the Pythagorean theorem: 1² + 1² = 2, so hypotenuse = √2).

Side ratios: leg : leg : hypotenuse = 1 : 1 : √2

If the leg is length a: hypotenuse = a√2
If the hypotenuse is length h: leg = h/√2 = h√2/2

Practical uses:
- Architecture and construction: a square cut diagonally creates two 45-45-90 triangles. The diagonal of a square with side s is s√2.
- Navigation: a bearing of 45° means equal north and east components.
- Physics: projectile at 45° maximises range (equal horizontal and vertical velocity components).
- Carpentry: 45° mitre cuts for picture frames and skirting boards.

Trig values from this triangle:
sin(45°) = cos(45°) = 1/√2 = √2/2 ≈ 0.7071
tan(45°) = cot(45°) = 1

°	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	Undef
120°	√3/2	−1/2	−√3
135°	√2/2	−√2/2	−1
150°	1/2	−√3/2	−1/√3
180°	0	−1	0
270°	−1	0	Undef
360°	0	1	0

NAME	FORMULA	RESULT
Pythagorean identity at 45°	sin²(45°)+cos²(45°) = (√2/2)²+(√2/2)² = 1/2+1/2	= 1 ✓
Tan from sin/cos	tan(45°) = sin(45°)/cos(45°) = (√2/2)/(√2/2)	= 1
Csc(45°)	csc(45°) = 1/sin(45°) = 1/(√2/2)	= √2
Sec(45°)	sec(45°) = 1/cos(45°) = 1/(√2/2)	= √2
Cot(45°)	cot(45°) = cos(45°)/sin(45°) = 1	= 1
45-45-90 triangle	Sides: 1, 1, √2 (ratio)	Both legs equal, hypotenuse = leg × √2
Co-function identity	sin(45°) = cos(45°) = √2/2	45° is self-complementary: 90° - 45° = 45°
Double angle (sin)	sin(90°) = 2·sin(45°)·cos(45°) = 2·(√2/2)·(√2/2)	= 2·(1/2) = 1 ✓

Sin Cos Tan of 45 Degrees — Exact Values

TRIG VALUES AT 45°

UNIT CIRCLE

FORMULAS & IDENTITIES

WORKED EXAMPLE

FREQUENTLY ASKED QUESTIONS

RELATED CALCULATORS

MORE TRIGONOMETRY CALCULATORS

Function	Exact Value	Decimal
sin(45°)	√2/2	0.70710678
cos(45°)	√2/2	0.70710678
tan(45°)	1	1
csc(45°)	√2	1.41421356
sec(45°)	√2	1.41421356
cot(45°)	1	1