Trig Expressions: Maximum and Minimum Values Explained

A trigonometric expression such as 3 sin x + 4 cos x rises and falls as x changes, and it always has a highest value and a lowest value. Finding the maximum and minimum of such expressions answers the question "how big can this get?"

This matters wherever waves and oscillations appear. An engineer designing a circuit needs the maximum voltage; a physicist studying a vibrating spring needs the maximum displacement.

Key Concepts

The foundation is that sin x and cos x each range between −1 and 1, so a sin x alone ranges between −|a| and |a|. A mixture combines into a single wave: a sin x + b cos x = R sin(x + α), where R = √(a² + b²) and α satisfies cos α = a/R, sin α = b/R. Since sin(x + α) ranges between −1 and 1, the whole expression ranges between −R and +R. For squared terms, sin²x ranges from 0 to 1, and the substitution t = sin x turns them into ordinary quadratics on −1 ≤ t ≤ 1.

Formulas

• Maximum of a sin x + b cos x = √(a² + b²); minimum = −√(a² + b²).
• sin²x and cos²x lie in [0, 1]; |sin x| lies in [0, 1].
• For a sin²x + b sin x + c, substitute t = sin x and find the max and min of at² + bt + c for −1 ≤ t ≤ 1, checking the vertex and both endpoints.

Worked Example

Find the maximum and minimum values of 3 sin x + 4 cos x.

1. Compute R = √(3² + 4²) = √(9 + 16) = √25 = 5.
2. Rewrite: 3 sin x + 4 cos x = 5 sin(x + α), where cos α = 3/5 and sin α = 4/5.
3. Multiplying the range [−1, 1] of sin(x + α) by 5 gives [−5, 5]: maximum 5, minimum −5.

A second pattern: for 2 sin²x + 3 sin x, put t = sin x (−1 ≤ t ≤ 1), giving f(t) = 2t² + 3t. The vertex at t = −3/4 gives f = −9/8; the endpoints give f(1) = 5 and f(−1) = −1. So the minimum is −9/8 and the maximum is 5.

Common Misconceptions

• "The maximum of a sin x + b cos x is a + b." Wrong. Here a + b = 7 but the true maximum is 5, because the two terms never peak at the same x. The answer is always √(a² + b²).
• "The minimum is 0." Wrong. The expression is symmetric: the minimum is the negative of the maximum. Here it is −5, not 0.
• "sin²x ranges from −1 to 1." Wrong. Squaring removes the sign, so sin²x ranges from 0 to 1. Using −1 as the lowest value ruins quadratic-type answers.

Real-World Applications

Electrical engineers use the √(a² + b²) rule to find peak voltage when two alternating signals combine, which fixes the rating of every component in the circuit. Physicists find the maximum stretch of an oscillating spring the same way. In signal processing, the maximum of a combined wave tells engineers how strong an antenna signal can get.

In short: a sin x + b cos x always lies between −√(a² + b²) and +√(a² + b²). Never add a and b, never assume the minimum is zero, and for squared terms substitute t = sin x and check the vertex and endpoints.