Ratios & Identities: Greatest & Least Values of Trigonometric Expressions

Understanding the greatest and least values of trigonometric expressions is crucial for solving a variety of problems in trigonometry. These concepts are often tested in exams and are applicable in many fields, including physics, engineering, and computer science.

Basic Trigonometric Ratios

Before diving into the greatest and least values, let's review the basic trigonometric ratios for an angle $\theta$ in a right-angled triangle:

• Sine ($\sin$): $\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}$
• Cosine ($\cos$): $\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}$
• Tangent ($\tan$): $\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{\sin(\theta)}{\cos(\theta)}$

Trigonometric Identities

Several identities are essential for finding the greatest and least values of trigonometric expressions:

• Pythagorean Identity: $\sin^2(\theta) + \cos^2(\theta) = 1$
• Reciprocal Identities:
  • $\csc(\theta) = \frac{1}{\sin(\theta)}$
  • $\sec(\theta) = \frac{1}{\cos(\theta)}$
  • $\cot(\theta) = \frac{1}{\tan(\theta)}$
• Double Angle Identities:
  • $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$
  • $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta)$
  • $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$

Greatest and Least Values

The greatest and least values of trigonometric functions occur at specific points within their domains. For the basic trigonometric functions:

• $\sin(\theta)$ and $\cos(\theta)$ have a range of $[-1, 1]$, meaning their greatest value is $1$ and their least value is $-1$.
• $\tan(\theta)$ has a range of $(-\infty, \infty)$, but it has no finite greatest or least value since it approaches infinity as $\theta$ approaches $\frac{\pi}{2}$ and $-\frac{\pi}{2}$.

For more complex trigonometric expressions, we often use trigonometric identities and calculus to find the greatest and least values.

Example 1: Greatest and Least Values of $\sin^2(\theta) + \cos^2(\theta)$

Using the Pythagorean Identity, we know that $\sin^2(\theta) + \cos^2(\theta) = 1$. Therefore, the greatest and least values of this expression are both $1$.

Example 2: Greatest and Least Values of $A\sin(\theta) + B\cos(\theta)$

For the expression $A\sin(\theta) + B\cos(\theta)$, where $A$ and $B$ are constants, we can find the greatest and least values by converting it into a single trigonometric function. This is done by using the following identity:

$$ A\sin(\theta) + B\cos(\theta) = R\sin(\theta + \phi) $$

where $R = \sqrt{A^2 + B^2}$ and $\phi$ is the phase angle such that $\cos(\phi) = \frac{A}{R}$ and $\sin(\phi) = \frac{B}{R}$. The greatest value of this expression is $R$, and the least value is $-R$.

Example 3: Greatest and Least Values of $\sin(\theta) + \sin(2\theta)$

To find the greatest and least values of $\sin(\theta) + \sin(2\theta)$, we can use the double angle identity for $\sin(2\theta)$:

$$ \sin(\theta) + \sin(2\theta) = \sin(\theta) + 2\sin(\theta)\cos(\theta) $$

This expression can be maximized and minimized by considering the ranges of $\sin(\theta)$ and $\cos(\theta)$. The greatest value occurs when both $\sin(\theta)$ and $\cos(\theta)$ are at their maximum, which is $1$. Thus, the greatest value is $1 + 2(1)(1) = 3$. The least value occurs when $\sin(\theta) = 1$ and $\cos(\theta) = -1$, giving us $1 + 2(1)(-1) = -1$.

Summary Table

Expression	Greatest Value	Least Value	Method Used
$\sin(\theta)$	$1$	$-1$	Range of $\sin$
$\cos(\theta)$	$1$	$-1$	Range of $\cos$
$\tan(\theta)$	Undefined	Undefined	Range of $\tan$
$\sin^2(\theta) + \cos^2(\theta)$	$1$	$1$	Pythagorean Identity
$A\sin(\theta) + B\cos(\theta)$	$\sqrt{A^2 + B^2}$	$-\sqrt{A^2 + B^2}$	Trigonometric Identity
$\sin(\theta) + \sin(2\theta)$	$3$	$-1$	Double Angle Identity

In conclusion, finding the greatest and least values of trigonometric expressions often involves using trigonometric identities and understanding the ranges of the trigonometric functions. These concepts are fundamental for solving complex problems in trigonometry and related fields.