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Simpler Form of Trigonometric Equations

Breakdown of complex expressions using trigonometric identities.

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Simpler Form of Trigonometric Equations

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

Trigonometric Equations

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.

  

 

Simplifying Expressions 

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


\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*}1+tan2xcsc2xsec2xcsc2x1cos2x1sin2x(1cos2x)(sin2x1)(1+tan2x=sec2x)Pythagorean Identity(sec2x=1cos2x and csc2x=1sin2x)Reciprocal Identity=(1cos2x)÷(1sin2x)=sin2xcos2x=tan2xQuotient Identity

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


\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*}sin2x+tan2x+cos2xsecx1+tan2xsecxsec2xsecx(sin2x+cos2x=1)Pythagorean Identity(1+tan2x=sec2x)Pythagorean Identity=secx

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


\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*}cosxcos3xcosx(1cos2x)Factor out cosx and sin2x=1cos2xcosx(sin2x)

Examples

Example 1

Earlier, you were asked to simplify \begin{align*}\cos^3 \theta = \frac{3\cos \theta + \cos 3\theta}{4}\end{align*}cos3θ=3cosθ+cos3θ4.

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

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

Substituting this into the original equation gives:

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

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

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

And finally pulling out a three and dividing:

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

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

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

Example 2

Simplify \begin{align*}\tan^3(x)\csc^3(x)\end{align*}tan3(x)csc3(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*}tan3(x)csc3(x)=sin3(x)cos3(x)×1sin3(x)=1cos3(x)=sec3(x)

Example 3

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

Start with \begin{align*}\sin^2(x) + \cos^2(x) = 1\end{align*}sin2(x)+cos2(x)=1, and divide everything through by \begin{align*}\sin^2(x)\end{align*}sin2(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*}sin2(x)+cos2(x)=1=sin2(x)sin2(x)+cos2(x)sin2(x)=1sin2(x)=1+cot2(x)=csc2(x)

Example 4

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


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

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

\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*}cot2(x)csc2(x)=cos2(x)sin2(x)1sin2(x)=cos2(x)

Review

Simplify each trigonometric expression as much as possible.

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

Review (Answers)

To see the Review answers, open this PDF file and look for section 3.3. 

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Vocabulary

Trigonometric Identity

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

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