Question

Solve each equation algebraically over the domain \(0^{\circ} \leq x<360^{\circ} .\) Verify your solution graphically. a) \(\cos x-\cos 2 x=0\) b) \(\sin ^{2} x-3 \sin x=4\) c) tan \(x \cos x \sin x-1=0\) d) \(\tan ^{2} x+\sqrt{3} \tan x=0\)

Short answer

a) 0°, 120°, 240°, 360°; b) 270°; c) 45°, 135°, 225°, 315°; d) 0°, 120°, 180°, 300°, 360°.

Step-by-step solution

Equation (a) - Simplifying Trigonometric Identity

Given equation: \ \ \ \(\text{cos}x - \text{cos}(2x) = 0\). \ Using the double angle formula, \ \ \(\text{cos}(2x) = 2\text{cos}^2(x) - 1\). \ Substituting in, we get: \ \ \(\text{cos}(x) - (2\text{cos}^2(x) -1) = 0\).

Equation (a) - Solving the Quadratic Equation

Simplify the equation: \ \ \(\text{cos}(x) - 2\text{cos}^2(x) + 1 = 0\). \ Rearrange to: \ \ \(2\text{cos}^2(x) - \text{cos}(x) - 1 = 0\). \ Factorize to: \ \ \((2\text{cos}(x) +1)(\text{cos}(x) -1) = 0\). \ This gives solutions: \ \ \(\text{cos}(x) = 1\) or \(\text{cos}(x) = -\frac{1}{2}\).

Equation (a) - Finding Angle Solutions

Solve for \(x\) in both cases: \ For \(\text{cos}(x) = 1\), \(x = 0^{\circ}, 360^{\circ}\). \ For \(\text{cos}(x) = -\frac{1}{2}\), \(x = 120^{\circ}, 240^{\circ}\).

Equation (b) - Simplifying Quadratic in Sine

Given equation: \ \ \(\text{sin}^2(x) - 3\text{sin}(x) = 4\). \ Rearrange and simplify to a standard quadratic form: \ \ \(\text{sin}^2(x) - 3\text{sin}(x) - 4 = 0\).

Equation (b) - Solving the Quadratic Equation

Factorize: \ \ \((\text{sin}(x) - 4)(\text{sin}(x) + 1) = 0\). \ This gives solutions: \ \ \(\text{sin}(x) = 4\) or \(\text{sin}(x) = -1\).

Equation (b) - Finding Angle Solutions

Solve for \(x\) in the valid range: \ \(\text{sin}(x) = 4\) is not valid (since sine value must be between -1 and 1). \ \(\text{sin}(x) = -1\) gives \(x = 270^{\circ}\).

Equation (c) - Simplifying and Solving

Given equation: \ \ \(\tan(x)\text{cos}(x)\text{sin}(x) - 1 = 0\). \ Using \(\tan(x) = \frac{\text{sin}(x)}{\text{cos}(x)}\): \ \ \(\frac{\text{sin}^2(x)}{\text{cos}(x)} - 1 = 0\). \ Multiply through by \(\text{cos}(x)\): \ \ \(\text{sin}^2(x) - \text{cos}(x) = 0\).

Equation (c) - Solving the Simplified Equation

Rearrange and simplify: \ \ \(\text{sin}^2(x) = \text{cos}(x)\). \ This implies \(\text{cos}(2x) = 2\text{cos}^2(x) - 1 = 0\). \ Solve \(\text{cos}(2x) = 0\): \ \(\frac{x}{2} = 90^{\circ}, 270^{\circ}\). \ Thus \(x = 45^{\circ}, 135^{\circ}, 225^{\circ}, 315^{\circ}\).

Equation (d) - Simplifying the Quadratic in Tangent

Given equation: \ \ \(\tan^2(x) + \sqrt{3}\tan(x) = 0\). Factorize: \ \ \(\tan(x)(\tan(x) + \sqrt{3}) = 0\). This gives solutions: \ \ \(\tan(x) = 0\) or \(\tan(x) = -\sqrt{3}\).

Equation (d) - Finding Angle Solutions

Solve for \(x\) in both cases: \ For \(\tan(x) = 0\): \ \(x = 0^{\circ}, 180^{\circ}, 360^{\circ}\). \ For \(\tan(x) = -\sqrt{3}\): \ \(x = 120^{\circ}, 300^{\circ}\).

Key concepts

cosine equation

To solve cosine equations, it's essential to understand the properties of the cosine function and its related identities. The cosine function, \[ \text{cos}(x) \], is defined for any angle, and its values range between -1 and 1. Common identities, such as the double angle formula \[ \text{cos}(2x) = 2\text{cos}^2(x) - 1 \], are crucial tools.

Let's break down the example given in the exercise: \[ \text{cos}(x) - \text{cos}(2x) = 0 \]. By substituting the double angle formula, we get \[ \text{cos}(x) - (2\text{cos}^2(x) - 1) = 0 \]. This simplifies to a quadratic equation in terms of \text{cos}(x): \[ 2\text{cos}^2(x) - \text{cos}(x) - 1 = 0 \].

Factoring gives: \[ (2\text{cos}(x) + 1)(\text{cos}(x) - 1) = 0 \]. Hence, \[ \text{cos}(x) = 1 \] or \[ \text{cos}(x) = -\frac{1}{2} \]. From trigonometric tables or the unit circle, \[ \text{cos}(x) = 1 \] when \[ x = 0^{\circ} \text{ or } 360^{\circ} \], and \[ \text{cos}(x) = -\frac{1}{2} \] when \[ x = 120^{\circ} \text{ or } 240^{\circ} \].
Understanding these properties helps in solving various cosine equations.

sine equation

Sine equations make use of the characteristics and identities of the sine function. The sine function, \[ \text{sin}(x) \], also ranges between -1 and 1. For solving quadratic equations involving sine as in \[ \text{sin}^2(x) - 3\text{sin}(x) = 4 \], the first step is to rearrange it into a standard form: \[ \text{sin}^2(x) - 3\text{sin}(x) - 4 = 0 \].

This quadratic equation can be factored: \[ (\text{sin}(x) - 4)(\text{sin}(x) + 1) = 0 \]. Setting each factor to zero, we get \[ \text{sin}(x) = 4 \text{ or } \text{sin}(x) = -1 \]. Since \[ \text{sin}(x) \] must lie between -1 and 1, \[ \text{sin}(x) = 4 \] is not a valid solution. Therefore, \[ \text{sin}(x) = -1 \] is the only solution, giving \[ x = 270^{\circ} \].

Recognizing such limitations in the function’s range is crucial for solving sine equations accurately.

tangent equation

Tangent equations often involve properties of the tangent function and its relation to sine and cosine. Tangent can be expressed as \[ \tan(x) = \frac{\text{sin}(x)}{\text{cos}(x)} \]. Consider the exercise equation: \[ \tan(x)\text{cos}(x)\text{sin}(x) - 1 = 0 \]. Substituting \[ \tan(x) = \frac{\text{sin}(x)}{\text{cos}(x)} \], we get a simplified form: \[ \frac{\text{sin}^2(x)}{\text{cos}(x)} - 1 = 0 \].

Multiplying through by \[ \text{cos}(x) \] yields: \[ \text{sin}^2(x) - \text{cos}(x) = 0 \], which implies \[ \text{cos}(2x) = 0 \]. Solving \[ \text{cos}(2x) = 0 \] gives \[ 2x = 90^{\circ}, 270^{\circ} \], hence \[ x = 45^{\circ}, 135^{\circ}, 225^{\circ}, 315^{\circ} \].

This method demonstrates how tangent equations can be approached through known identities.

trigonometric identities

Trigonometric identities are essential tools for solving trigonometric equations. These identities express the relationships between different trigonometric functions and simplify complex equations. Some key identities are:

- Pythagorean identities: \[ \text{sin}^2(x) + \text{cos}^2(x) = 1 \]

- Double angle formulas: \[ \text{cos}(2x) = 2\text{cos}^2(x) - 1 \text{ and } \text{sin}(2x) = 2\text{sin}(x)\text{cos}(x) \]

- Sum and difference formulas: \[ \text{cos}(a \text{±} b) = \text{cos}(a)\text{cos}(b) \text{∓} \text{sin}(a)\text{sin}(b) \]

These identities play a critical role, as seen in the original exercise where the double angle formula \[ \text{cos}(2x) = 2\text{cos}^2(x) - 1 \] simplifies the equation \[ \text{cos}(x) - \text{cos}(2x) = 0 \].

Learning and applying these foundational identities makes solving trigonometric equations more manageable and effective.