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Chapter 7: Problem 21

Simplify and check using a graphing calculator. $$\frac{4 \tan t \sec t+2 \sec t}{6 \tan t \sec t+2 \sec t}$$

Short Answer

Expert verified
The simplified expression is \( \frac{2 \ {tan t} + 1}{3 \ {tan t} + 1} \). Verify using a graphing calculator by confirming both expressions produce the same graph.

Step by step solution

01

Identify Common Factors

First, observe that both the numerator and the denominator share common factors. Rewrite each term to make these common factors more evident.Numerator: \[ 4 \tan t \ {sec t} + 2 \ {sec t} = 2 \ {sec t}(2 \ {tan t} + 1) \]Denominator: \[ 6 \ {tan t} \ {sec t} + 2 \ {sec t} = 2 \ {sec t}(3 \ {tan t} + 1) \]
02

Factor out the Common Terms

Next, factor out the common terms in the numerator and the denominator.\[ \frac{2 \ {sec t}(2 \ {tan t} + 1)}{2 \ {sec t}(3 \ {tan t} + 1)} \]
03

Simplify the Fraction

Cancel out the common terms present in the numerator and the denominator.\[ \frac{2 \ {sec t}(2 \ {tan t} + 1)}{2 \ {sec t}(3 \ {tan t} + 1)} = \frac{2 \ {tan t} + 1}{3 \ {tan t} + 1} \]
04

Verify Using a Graphing Calculator

To check the answer, use a graphing calculator to graph both the original expression and the simplified expression. They should produce the same graph. Enter the original expression: \[ y = \frac{4 \ {tan t} \ {sec t} + 2 \ {sec t}}{6 \ {tan t} \ {sec t} + 2 \ {sec t}} \]Now, enter the simplified expression:\[ y = \frac{2 \ {tan t} + 1}{3 \ {tan t} + 1} \]Check if the graphs overlap completely. If they do, the simplified expression is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Identities
Understanding trigonometric identities is key to simplifying expressions in trigonometry. Trigonometric identities are equations that hold true for all values of the variables involved. These include fundamental identities like:
  • \(\tan t = \frac{\text{sin} t}{\text{cos} t} \)
  • \(\text{sec} t = \frac{1}{\text{cos} t} \)
  • The Pythagorean identity: \(\text{sin}^2 t + \text{cos}^2 t = 1 \)
By recognizing and applying these identities, you can simplify complex trigonometric expressions. In our exercise, understanding that \(\text{sec} t = \frac{1}{\text{cos} t} \) helped identify common factors to simplify the expressions.
This is a fundamental skill in trigonometry and becomes easier with practice.
Factoring
Factoring involves breaking down an expression into products of simpler expressions that multiply together to give the original expression. This is crucial for simplifying fractions. In our given problem, the numerator and the denominator share common factors: \(\text{sec} t \). We rewrite each term to make these factors more obvious:
  • Numerator: \(\text{4 tan t sec t + 2 sec t = 2 sec t( 2 tan t + 1 )} \)
  • Denominator: \(\text{6 tan t sec t + 2 sec t = 2 sec t( 3 tan t + 1 )} \)
Next, by factoring out \(\text{2 sec t} \), we simplify the expression easily.
Thus, it becomes \(\frac{2 sec t(2 tan t + 1)}{2 sec t(3 tan t + 1)} \), which further simplifies to \(\frac{2 tan t + 1}{3 tan t + 1} \) after canceling \(\text{2 sec t} \). This step is crucial for straightforward simplification and shows the elegance of trigonometric factoring.
Using a Graphing Calculator
To verify mathematical simplifications, using a graphing calculator can be very helpful. It allows you to visually confirm that the original and simplified expressions produce the same results. Here's how you can use it for our problem:
  • Firstly, enter the original expression: \(\frac{4 \text{tan} t \text{sec} t + 2 \text{sec} t}{6 \text{tan} t \text{sec} t + 2 \text{sec} t} \)
  • Secondly, enter the simplified expression: \(\frac{2 \text{tan} t + 1}{3 \text{tan} t + 1} \)
  • Graph both expressions.
If the graphs of both equations overlap completely, your simplified expression is correct. This visual confirmation is a powerful tool to ensure accuracy in trigonometric simplifications. It also enhances understanding by providing a different perspective on the problem.

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Most popular questions from this chapter

Angles Between Lines. One of the identities gives an easy way to find an angle formed by two lines. Consider two lines with equations \(l_{1}: y=m_{1} x+b_{1}\) and \(l_{2}: y=m_{2} x+b_{2}\) (GRAPH CANNOT COPY) The slopes \(m_{1}\) and \(m_{2}\) are the tangents of the angles \(\theta_{1}\) and \(\theta_{2}\) that the lines form with the positive direction of the \(x\) -axis. Thus we have \(m_{1}=\tan \theta_{1}\) and \(m_{2}=\tan \theta_{2} .\) To find the measure of \(\theta_{2}-\theta_{1},\) or \(\phi,\) we proceed as follows: This formula also holds when the lines are taken in the reverse order. When \(\phi\) is acute, tan \(\phi\) will be positive. When \(\phi\) is obtuse, tan \(\phi\) will be negative. Find the measure of the angle from \(l_{1}\) to \(l_{2}\) $$\begin{aligned} &l_{1}: 2 x=3-2 y\\\ &l_{2}: x+y=5 \end{aligned}$$

Find the following. \(\tan \left(\frac{1}{2} \sin ^{-1} \frac{1}{2}\right)\)

The acceleration due to gravity is often denoted by \(g\) in a formula such as \(S=\frac{1}{2} g t^{2},\) where \(S\) is the distance that an object falls in time \(t .\) The number \(g\) relates to motion near the earth's surface and is generally considered constant. In fact, however, \(g\) is not constant, but varies slightly with latitude. Latitude is used to measure north-south location on the earth between the equator and the poles. If \(\phi\) stands for latitude, in degrees, \(g\) is given with good approximation by the formula \(g=9.78049\left(1+0.005288 \sin ^{2} \phi-0.000006 \sin ^{2} 2 \phi\right)\), where \(g\) is measured in meters per second per second at sea level. a) Chicago has latitude \(42^{\circ} \mathrm{N}\). Find \(g .\) b) Philadelphia has latitude \(40^{\circ} \mathrm{N}\). Find \(g\). c) Express \(g\) in terms of \(\sin \phi\) only. That is, eliminate the double angle.

Prove the identity. $$\ln |\tan x|=-\ln |\cot x|$$

Fill in the blank with the correct term. Some of the given choices will not be used. $$\begin{array}{ll}\text { linear speed } & \text { congruent } \\ \text { angular speed } & \text { circular } \\ \text { angle of elevation } & \text { periodic } \\ \text { angle of depression } & \text { period } \\ \text { complementary } & \text { amplitude } \\ \text { supplementary } & \text { quadrantal } \\ \text { similar } & \text { radian measure }\end{array}$$ Two positive angles are ____________ if their sum is \(180^{\circ}\).
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