9.5 Solving Trigonometric Equations

Trigonometric equations are a key part of understanding how angles and ratios work in triangles and circles. These equations help us solve real-world problems in fields like engineering and physics, where we need to figure out distances or angles.

Solving these equations involves isolating trig functions, using identities, and applying algebraic techniques. We'll learn how to handle basic equations, quadratic forms, and multiple angle problems. Understanding these methods will give us powerful tools for tackling complex geometric challenges.

Solving Trigonometric Equations

Solving basic trigonometric equations

• Isolate the trigonometric function on one side of the equation by using algebraic operations (addition, subtraction, multiplication, division)
• Determine the angle(s) that satisfy the equation
  • For $\sin \theta = a$, solutions are $\theta = \arcsin a + 2\pi n$ or $\theta = \pi - \arcsin a + 2\pi n$ where $n$ is an integer
  • For $\cos \theta = a$, solutions are $\theta = \arccos a + 2\pi n$ or $\theta = -\arccos a + 2\pi n$ where $n$ is an integer
• Consider the domain and range of the trigonometric functions to determine the number of solutions based on the given interval or context
• Use reference angles to find solutions in different quadrants of the unit circle

Algebraic techniques for trigonometric equations

• Manipulate the equation to isolate the trigonometric function by applying algebraic operations and properties of equality to maintain the equation's balance
• Solve the resulting equation using the techniques for basic trigonometric equations outlined in the previous section
• Example: For $2\sin \theta + 1 = 0$, subtract 1 from both sides and divide by 2 to isolate $\sin \theta$, then solve using the basic techniques

Calculator use in trigonometric solutions

• Isolate the trigonometric function on one side of the equation using algebraic techniques
• Use the inverse trigonometric functions on a calculator to find the angle(s) that satisfy the equation
  • For $\sin \theta = a$, use $\theta = \sin^{-1} a$
  • For $\cos \theta = a$, use $\theta = \cos^{-1} a$
• Consider the domain and range of the trigonometric functions to determine additional solutions beyond the calculator's output, which typically gives solutions in the interval $[0, 2\pi]$ or $[-\pi, \pi]$

Quadratic-form trigonometric equations

• Recognize equations in the form $a\sin^2 \theta + b\sin \theta + c = 0$ or $a\cos^2 \theta + b\cos \theta + c = 0$
• Substitute $u = \sin \theta$ or $u = \cos \theta$ to transform the equation into a quadratic equation in terms of $u$
• Solve the quadratic equation for $u$ using factoring, completing the square, or the quadratic formula
• Substitute the solutions for $u$ back into the original trigonometric equation and solve for $\theta$ using basic trigonometric equation techniques
• Example: For $2\sin^2 \theta - 3\sin \theta - 2 = 0$, substitute $u = \sin \theta$ to get $2u^2 - 3u - 2 = 0$, solve for $u$, then solve for $\theta$

Fundamental trigonometric identities in equations

• Recognize and apply trigonometric identities to simplify the equation
  • Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$
  • Double angle formulas: $\sin 2\theta = 2\sin \theta \cos \theta$ and $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$
  • Half-angle formulas: $\sin^2 \theta = \frac{1 - \cos 2\theta}{2}$ and $\cos^2 \theta = \frac{1 + \cos 2\theta}{2}$
• Solve the simplified equation using appropriate techniques based on the form of the resulting equation
• Example: For $\sin^2 \theta + \cos^2 \theta = \frac{1}{2}$, apply the Pythagorean identity to simplify to $1 = \frac{1}{2}$, which has no solution

Multiple angle trigonometric equations

• Break down compound expressions into simpler trigonometric functions using trigonometric identities
  • Sum and difference formulas: $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ and $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$
  • Product-to-sum formulas: $\sin A \sin B = \frac{\cos(A - B) - \cos(A + B)}{2}$ and $\cos A \cos B = \frac{\cos(A - B) + \cos(A + B)}{2}$
• Solve the resulting equations using appropriate techniques based on the form of the simplified equation
• Example: For $\sin 2\theta = \frac{1}{2}$, use the double angle formula to rewrite as $2\sin \theta \cos \theta = \frac{1}{2}$, then solve using algebraic techniques

Right triangle trigonometry applications

• Identify the given information and the unknown quantity in the problem, labeling them in a sketched diagram of the right triangle
• Set up trigonometric equations using the relationships between sides and angles in a right triangle
  • $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
• Solve the trigonometric equations to find the unknown quantity, which may be a side length or an angle measure
• Interpret the solution in the context of the problem, ensuring that the answer makes sense given the problem's constraints
• Example: Given a right triangle with hypotenuse 10 and an angle of 30°, find the opposite side using $\sin 30° = \frac{\text{opposite}}{10}$ and solving for the opposite side length

Additional Trigonometric Functions and Concepts

• Understand the relationships between trigonometric ratios and their reciprocals
  • Cotangent (cot) is the reciprocal of tangent
  • Secant (sec) is the reciprocal of cosine
  • Cosecant (csc) is the reciprocal of sine
• Recognize the periodicity of trigonometric functions and how it affects solution sets
• Apply the concept of the unit circle to visualize and solve trigonometric equations