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# 3.3: Simpler Form of Trigonometric Equations

Difficulty Level: At Grade Created by: CK-12
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Practice Simpler Form of Trigonometric Equations

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Sometimes things are simpler than they look. For example, trigonometric identities can sometimes be reduced to simpler forms by applying other rules. For example, can you find a way to simplify

cos3θ=3cosθ+cos3θ4\begin{align*}\cos^3 \theta = \frac{3\cos \theta + \cos 3\theta}{4}\end{align*}

Keep reading, and during this Concept you'll learn ways to break down complex trigonometric equations into simpler forms. You'll be able to apply this information to the equation above.

### Watch This

In the first part of this video, you'll learn how to use trigonometric substitution to simplify equations.

### Guidance

By this time in your school career you have probably seen trigonometric functions represented in many ways: ratios between the side lengths of right triangles, as functions of coordinates as one travels along the unit circle and as abstract functions with graphs. Now it is time to make use of the properties of the trigonometric functions to gain knowledge of the connections between the functions themselves. The patterns of these connections can be applied to simplify trigonometric expressions and to solve trigonometric equations.

In order to do this, look for parts of the complex trigonometric expression that might be reduced to fewer trig functions if one of the identities you already know is applied to the expression. As you apply identities, some complex trig expressions have parts that can be cancelled out, others can be reduced to fewer trig functions. Observe how this is accomplished in the examples below.

#### Example A

Simplify the following expression using the basic trigonometric identities: 1+tan2xcsc2x\begin{align*}\frac{1 + \tan^2 x}{\csc^2 x}\end{align*}

Solution: 1+tan2xcsc2xsec2xcsc2x1cos2x1sin2x(1cos2x)(sin2x1)(1+tan2x=sec2x)Pythagorean Identity(sec2x=1cos2x and csc2x=1sin2x)Reciprocal Identity=(1cos2x)÷(1sin2x)=sin2xcos2x=tan2xQuotient Identity\begin{align*}\frac{1 + \tan^2 x}{\csc^2 x}&\ldots( 1 + \tan^2 x = \sec^2 x ) \text{Pythagorean Identity} \\ \frac{\sec^2 x}{\csc^2 x} & \ldots (\sec^2 x = \frac{1}{\cos^2 x}\ \text{and}\ \csc^2 x = \frac{1}{\sin^2 x}) \text{Reciprocal Identity} \\ \frac{\frac{1}{\cos^2 x}}{\frac{1}{\sin^2 x}} &= \left (\frac{1}{\cos^2 x} \right ) \div \left (\frac{1}{\sin^2 x} \right ) \\ \left (\frac{1}{\cos^2 x} \right ) \cdot \left (\frac{\sin^2 x}{1} \right ) &= \frac{\sin^2 x}{\cos^2 x}\\ & = \tan^2 x \rightarrow \text{Quotient Identity}\end{align*}

#### Example B

Simplify the following expression using the basic trigonometric identities: sin2x+tan2x+cos2xsecx\begin{align*}\frac{\sin^2 x + \tan^2 x + \cos^2 x}{\sec x}\end{align*}

Solution: sin2x+tan2x+cos2xsecx1+tan2xsecxsec2xsecx(sin2x+cos2x=1)Pythagorean Identity(1+tan2x=sec2x)Pythagorean Identity=secx\begin{align*} \frac{\sin^2 x + \tan^2 x + \cos^2 x}{\sec x} &\ldots (\sin^2 x + \cos^2 x = 1) \text{Pythagorean Identity} \\ \frac{1 + \tan^2 x}{\sec x} & \ldots (1 + \tan^2 x = \sec^2 x) \text{Pythagorean Identity} \\ \frac{\sec^2 x}{\sec x} & = \sec x\end{align*}

#### Example C

Simplify the following expression using the basic trigonometric identities: cosxcos3x\begin{align*}\cos x - \cos^3x\end{align*}

Solution: cosxcos3xcosx(1cos2x)Factor out cosx and sin2x=1cos2xcosx(sin2x)\begin{align*}& \cos x - \cos^3 x \\ & \cos x (1 - \cos^2 x) \qquad \ldots \text{Factor out}\ \cos x \ \text{and}\ \sin^2 x = 1 - \cos^2 x \\ & \cos x (\sin^2 x)\end{align*}

### Vocabulary

Trigonometric Identity: A trigonometric identity is an expression which relates one trig function on the left side of an equals sign to another trig function on the right side of the equals sign.

### Guided Practice

1. Simplify tan3(x)csc3(x)\begin{align*}\tan^3(x)\csc^3(x)\end{align*}

2. Show that cot2(x)+1=csc2(x)\begin{align*}\cot^2(x) + 1 = \csc^2(x)\end{align*}

3. Simplify csc2(x)1csc2(x)\begin{align*}\frac{\csc^2(x)-1}{\csc^2(x)}\end{align*}

Solutions:

1. tan3(x)csc3(x)=sin3(x)cos3(x)×1sin3(x)=1cos3(x)=sec3(x)\begin{align*} \tan^3(x)\csc^3(x)\\ =\frac{\sin^3(x)}{\cos^3(x)} \times \frac{1}{\sin^3(x)}\\ =\frac{1}{\cos^3(x)}\\ =\sec^3(x) \end{align*}

2. Start with sin2(x)+cos2(x)=1\begin{align*}\sin^2(x) + \cos^2(x) = 1\end{align*}, and divide everything through by sin2(x)\begin{align*}\sin^2(x)\end{align*}:

sin2(x)+cos2(x)=1=sin2(x)sin2(x)+cos2(x)sin2(x)=1sin2(x)=1+cot2(x)=csc2(x)\begin{align*} \sin^2(x) + \cos^2(x) = 1\\ =\frac{\sin^2(x)}{\sin^2(x)} + \frac{\cos^2(x)}{\sin^2(x)} = \frac{1}{\sin^2(x)}\\ =1 + \cot^2(x) = \csc^2(x)\\ \end{align*}

3. csc2(x)1csc2(x)\begin{align*} \frac{\csc^2(x)-1}{\csc^2(x)}\\ \end{align*}

Using cot2(x)+1=csc2(x)\begin{align*}\cot^2(x) + 1 = \csc^2(x)\end{align*} that was proven in #2, you can find the relationship: cot2(x)=csc2(x)1\begin{align*}\cot^2(x) = \csc^2(x)-1\end{align*}, you can substitute into the above expression to get:

cot2(x)csc2(x)=cos2(x)sin2(x)1sin2(x)=cos2(x)\begin{align*} \frac{\cot^2(x)}{\csc^2(x)}\\ =\frac{\frac{\cos^2(x)}{\sin^2(x)}}{\frac{1}{\sin^2(x)}}\\ =\cos^2(x)\\ \end{align*}

### Concept Problem Solution

The original problem is to simplify

cos3θ=3cosθ+cos3θ4\begin{align*}\cos^3 \theta = \frac{3\cos \theta + \cos 3\theta}{4}\end{align*}

The easiest way to start is to recognize the triple angle identity:

cos3θ=cos3θ3sin2θcosθ\begin{align*}\cos 3\theta = \cos^3 \theta - 3\sin^2 \theta \cos \theta\end{align*}

Substituting this into the original equation gives:

cos3θ=3cosθ+(cos3θ3sin2θcosθ)4\begin{align*}\cos^3 \theta = \frac{3\cos \theta + (\cos^3 \theta - 3\sin^2 \theta \cos \theta)}{4}\end{align*}

Notice that you can then multiply by four and subtract a cos3θ\begin{align*}\cos^3 \theta\end{align*} term:

3cos3θ=3cosθ3sin2θcosθ\begin{align*}3 \cos^3 \theta = 3 \cos \theta - 3 \sin^2 \theta \cos \theta\end{align*}

And finally pulling out a three and dividing:

cos3θ=cosθsin2θcosθ\begin{align*}\cos^3 \theta = \cos \theta - \sin^2 \theta \cos \theta\end{align*}

Then pulling out a cosθ\begin{align*}\cos \theta\end{align*} and dividing:

cos2θ=1sin2θ\begin{align*}\cos^2 \theta = 1 - \sin^2 \theta\end{align*}

### Practice

Simplify each trigonometric expression as much as possible.

1. sin(x)cot(x)\begin{align*}\sin(x)\cot(x)\end{align*}
2. cos(x)tan(x)\begin{align*}\cos(x)\tan(x)\end{align*}
3. 1+tan(x)1+cot(x)\begin{align*}\frac{1+\tan(x)}{1+\cot(x)}\end{align*}
4. 1sin2(x)1+sin(x)\begin{align*}\frac{1-\sin^2(x)}{1+\sin(x)}\end{align*}
5. sin2(x)1+cos(x)\begin{align*}\frac{\sin^2(x)}{1+\cos(x)}\end{align*}
6. (1+tan2(x))(sec2(x))\begin{align*}(1+\tan^2(x))(\sec^2(x))\end{align*}
7. sin(x)(tan(x)+cot(x))\begin{align*}\sin(x)(\tan(x)+\cot(x))\end{align*}
8. sec(x)sin(x)sin(x)cos(x)\begin{align*}\frac{\sec(x)}{\sin(x)}-\frac{\sin(x)}{\cos(x)}\end{align*}
9. sin(x)cot2(x)sin(x)cos2(x)\begin{align*}\frac{\sin(x)}{\cot^2(x)}-\frac{\sin(x)}{\cos^2(x)}\end{align*}
10. 1+sin(x)cos(x)sec(x)\begin{align*}\frac{1+\sin(x)}{\cos(x)}-\sec(x)\end{align*}
11. sin2(x)sin4(x)cos2(x)\begin{align*}\frac{\sin^2(x)-\sin^4(x)}{\cos^2(x)}\end{align*}
12. tan(x)csc2(x)+tan(x)sec2(x)\begin{align*}\frac{\tan(x)}{\csc^2(x)}+\frac{\tan(x)}{\sec^2(x)}\end{align*}
13. \begin{align*}\sqrt{1-cos^2(x)}\end{align*}
14. \begin{align*}(1-\sin^2(x))(\cos(x))\end{align*}
15. \begin{align*}(\sec^2(x)+\csc^2(x))-(\tan^2(x)+\cot^2(x))\end{align*}

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### Vocabulary Language: English

Trigonometric Identity

A trigonometric identity is an equation that relates two or more trigonometric functions.

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