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Trigonometric equationsTrigonometryHigh schoolabout 1:42

Solve cos(x) = sqrt(2)/2: animated lesson

A narrated, step-by-step trigonometric equations video lesson with captions. Final answer: x = π/4 + 2πk, x = 7π/4 + 2πk (k any integer).

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Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.6 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Solving an equation
$\cos\left( x\right) = \frac{\sqrt{2}}{2}$
2. Steps, part 1
1. $\cos\left( x\right) = \frac{\sqrt{2}}{2}$
  Start with the equation
2. $f(x) = \cos\left( x\right)-0.7071067811865476 = 0$
  Rewrite as f(x) = 0
3. The left side repeats every 2π, so find the solutions in one period
4. $x = \frac{\pi}{4},\ x = \frac{7\pi}{4}$
  Solutions in one period [0, 2π)
3. Steps, part 2
1. $x = \frac{\pi}{4} + 2\pi k,\quad x = \frac{7\pi}{4} + 2\pi k,\quad k \in \mathbb{Z}$
  Add any whole number of periods
4. The answer
$x = \frac{\pi}{4} + 2\pi k,\ x = \frac{7\pi}{4} + 2\pi k,\ k \in \mathbb{Z}$
x = π/4 + 2πk, x = 7π/4 + 2πk (k any integer)
5. Check
- x = 0.78539816: left side = 0.70710678, right side = 0.70710678 ✓
- x = 5.4977871: left side = 0.70710678, right side = 0.70710678 ✓

Transcript

The problem. Here is the problem: cos(x) equals sqrt(2)/2. A periodic equation in x: it has infinitely many solutions, one family for each solution in a single period.

Steps, part 1. Start with the equation. Rewrite as f(x) equals 0. The left side repeats every 2pi, so find the solutions in one period. f(x plus 2pi) equals f(x) for every x, so every solution is one of the solutions in [0, 2pi) plus a whole number of periods. Solutions in one period [0, 2pi).

Steps, part 2. Add any whole number of periods.

The answer. So the answer is x equals pi/4 plus 2pik, x equals 7pi/4 plus 2pik (k any integer). Each solution in one period was found numerically, matched to an exact value and confirmed by substitution; adding whole periods gives all solutions.

Check. x equals 0.78539816: left side equals 0.70710678, right side equals 0.70710678, which checks out. x equals 5.4977871: left side equals 0.70710678, right side equals 0.70710678, which checks out.

About trigonometric equations

Solve trigonometric equations and give every solution.

Solve cos(x) = sqrt(2)/2: Animated Step-by-Step Lesson