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Trigonometry Angles — Pi/4 (45°) — Definition, Formula & Examples

Pi/4 (45°) is one of the standard angles in trigonometry where all six trig values can be expressed as exact fractions involving √2. At this angle, sine and cosine are equal, both equaling √2/2, and tangent equals 1.

For the angle θ=π4\theta = \frac{\pi}{4} radians (equivalently 45°), the trigonometric ratios are derived from an isosceles right triangle with legs of length 1 and hypotenuse 2\sqrt{2}: sin⁡π4=cos⁡π4=22\sin\frac{\pi}{4} = \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}, tan⁡π4=1\tan\frac{\pi}{4} = 1, csc⁡π4=sec⁡π4=2\csc\frac{\pi}{4} = \sec\frac{\pi}{4} = \sqrt{2}, and cot⁡π4=1\cot\frac{\pi}{4} = 1.

Key Formula

sin⁡π4=cos⁡π4=22,tan⁡π4=1\sin\frac{\pi}{4} = \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}, \quad \tan\frac{\pi}{4} = 1
Where:
  • π4\frac{\pi}{4} = The angle, equal to 45 degrees
  • 22\frac{\sqrt{2}}{2} = Exact value, approximately 0.7071

How It Works

Draw an isosceles right triangle with two 45° angles and two legs of equal length. If each leg has length 1, the Pythagorean theorem gives a hypotenuse of 12+12=2\sqrt{1^2 + 1^2} = \sqrt{2}. Sine is opposite over hypotenuse: 12=22\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}. Cosine is adjacent over hypotenuse, which gives the same value since both legs are equal. Tangent is opposite over adjacent: 11=1\frac{1}{1} = 1. The reciprocal functions follow directly.

Worked Example

Problem: Find the exact length of the hypotenuse of a right triangle with a 45° angle and legs of length 5.
Step 1: In a 45-45-90 triangle, both legs are equal. Use cosine to relate the leg to the hypotenuse.
cos⁡45°=adjacenthypotenuse=5h\cos 45° = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{5}{h}
Step 2: Substitute the exact value of cos 45° and solve for h.
22=5h  ⟹  h=5⋅22=102=52\frac{\sqrt{2}}{2} = \frac{5}{h} \implies h = \frac{5 \cdot 2}{\sqrt{2}} = \frac{10}{\sqrt{2}} = 5\sqrt{2}
Answer: The hypotenuse is 52≈7.0715\sqrt{2} \approx 7.071.

Why It Matters

The 45° angle appears constantly in physics (projectile motion at maximum range), engineering (diagonal bracing), and computer graphics (rotation matrices). Knowing these exact values by heart eliminates calculator dependence on precalculus exams and speeds up work in calculus when evaluating integrals and derivatives of trig functions.

Common Mistakes

Mistake: Writing sin⁡45°=12\sin 45° = \frac{1}{2} by confusing it with 30°.
Correction: At 45°, sine and cosine are both 22\frac{\sqrt{2}}{2}, not 12\frac{1}{2}. The value 12\frac{1}{2} belongs to sin⁡30°\sin 30°.