Solving cos x = x with two graphs

The problem

Try solving cos x = x by algebra. You cannot get x alone: one x sits inside a cosine and the other does not. No rearranging helps.

So draw instead. Plot y = cos x and y = x on the same axes. Where the two graphs cross, both sides are equal, and that x is the solution.

To solve f(x) = g(x) graphically, draw y = f(x) and y = g(x) on the same axes. The x-coordinates of the intersection points are the solutions. For cos x = x there is exactly one: x ≈ 0.739 radian.

Key terms

Crossing = solution

At a crossing, both graphs have the same y for the same x, so f(x) = g(x) there.

Radians

The line y = x compares an angle with a plain number, so x must be in radians. In degrees, x = 43 would never equal a cosine.

Scale

The book uses π/6 per small square across and 0.1 up. With unequal scales the line y = x is not at 45°.

Check by substitution

Put your reading into both sides. If they nearly agree, the reading is good; if not, zoom in.

Short questions with model answers

1. Q1. Solve cos x = x graphically for −π ≤ x ≤ π.

   • cos 0 = 1, cos(±π/6) = 0.87, cos(±π/3) = 0.5, cos(±π/2) = 0, cos(±2π/3) = −0.5, cos(±π) = −1
   • x = π/6: 0.87 > 0.52; x = π/3: 0.5 < 1.05
   • x ≈ 0.74 rad

   x ≈ 0.739 rad

2. Q2. Check the reading x ≈ 0.739, and the printed value 43π/180.

   • cos 0.739 = 0.7391 ≈ 0.739 ✓
   • 43π/180 = 0.7505; cos 0.7505 = 0.7314 ≠ 0.7505

   0.739 checks; 43π/180 = 0.7505 is slightly too big

3. Q3. Solve sin x = x graphically.

   • sin 0 = 0 ⇒ (0, 0)
   • x = 0.5: sin 0.5 = 0.479 < 0.5; x = 1: 0.841 < 1

   x = 0

Common mistakes

• ✗ Marking the x-axis in degrees and drawing y = x on it.

  ✓ y = x needs x as a number, so use radians: π/6 = 0.524, π/3 = 1.047. Convert the final answer to degrees only if asked.

• ✗ Writing 43π/180 = 0.73 as the answer.

  ✓ 43π/180 = 0.7505. The crossing is at 0.739 rad; check it: cos 0.739 = 0.7391.

• ✗ Plotting cos(−π/3) = −0.5.

  ✓ Cosine is even: cos(−π/3) = cos(π/3) = 0.5. A wrong sign here bends the curve the wrong way.

MCQs

1. 1. How many real solutions does cos x = x have?

   1. (a) 0
   2. (b) 1
   3. (c) 2
   4. (d) ∞
   Show answer

   (b) The graphs cross once, near 0.739; for x < 0, cos x > 0 > x, and for x > 1, x > 1 ≥ cos x.

2. 2. The solution of cos x = x lies between:

   1. (a) 0 and π/6
   2. (b) π/6 and π/3
   3. (c) π/3 and π/2
   4. (d) π/2 and π
   Show answer

   (b) At π/6, cos x = 0.87 > 0.52; at π/3, cos x = 0.5 < 1.05. The sign of cos x − x changes in between.

3. 3. 43π/180 radian equals about:

   1. (a) 0.73
   2. (b) 0.75
   3. (c) 0.43
   4. (d) 1.35
   Show answer

   (b) 43 × 3.1416/180 = 0.7505.

4. 4. The only real solution of sin x = x is:

   1. (a) x = 0
   2. (b) x = π/2
   3. (c) x = 1
   4. (d) x = π
   Show answer

   (a) The graphs meet only at the origin; for x ≠ 0, |sin x| < |x|.

5. 5. cos(−π/3) equals:

   1. (a) −0.5
   2. (b) 0.5
   3. (c) 0.87
   4. (d) −0.87
   Show answer

   (b) Cosine is even, so cos(−π/3) = cos(π/3) = 0.5.

Quick revision

• To solve f(x) = g(x), draw both graphs on the same axes; the x-values where they cross are the solutions.
• cos x = x has exactly one solution, x ≈ 0.739 rad; sin x = x has only x = 0.
• Work in radians, and always check a reading by putting it back into both sides.