Solve the given trigonometry equation

In summary, the conversation discusses the repetition of tangent on a cycle of ##π## radians and the use of 2 in the formula. The speaker questions the necessity of this and prefers using ##π## radians. They also mention a known form for ##\tan^{-1}## and explain how it can be used to find desired values. They provide three equations with the variable x and a range for the solution.
  • #1
chwala
Gold Member
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Homework Statement
see attached
Relevant Equations
Trigonometry
This is the problem. The question is simple i just need some clarification as indicated on the part highlighted below in red.

1662037532264.png
Now from my understanding tangent repeats on a cycle of ##π## radians...why do we have 2 the part circled in red below i.e ##2##? This is the part that i need clarity. Why consider cycles for sine and cosine here?
It's straightforward and less stressful to just use ##π##radians in my opinion.

1662037959983.png


The other form given below is the one that i know and am very much conversant with;

1662037901886.png
 
Last edited:
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  • #2
We know that ##\tan^{-1}####\left[-\dfrac{5}{9}\right]=-0.5071##.

Tangent is negative in the second and fourth quadrant respectively. This will realize the given values: ##-0.5071+π## and ##2π-0.5071## respectively... giving us the desired ##2.6345## and ##5.7761##
i.e
##2x=-0.5071+nπ⇒x=-0.25355+\dfrac{1}{2}nπ##

##2x=2.6345+nπ⇒x=1.31725+\dfrac{1}{2}nπ##

##2x=5.7761+nπ⇒x=2.88805+\dfrac{1}{2}nπ##

where ##n=0, ±1, ±2...## and given that the solution lies on ##-10≤x≤0## then we shall get the answers as shown on the text.

Cheers.
 
Last edited:

Related to Solve the given trigonometry equation

1. How do I solve a trigonometry equation?

To solve a trigonometry equation, you will need to use trigonometric identities and properties to simplify the equation. Then, you can use algebraic methods such as factoring or the quadratic formula to solve for the unknown variable.

2. What are the basic trigonometric identities?

The basic trigonometric identities include the Pythagorean identities, reciprocal identities, quotient identities, and co-function identities. These identities can be used to simplify trigonometric expressions and equations.

3. Can I use a calculator to solve a trigonometry equation?

Yes, you can use a calculator to solve a trigonometry equation. However, it is important to understand the steps and methods used to solve the equation by hand in order to verify the accuracy of the calculator's results.

4. How do I know if I need to use the unit circle to solve a trigonometry equation?

If the equation involves angles that are not commonly found on the unit circle (such as 15 or 75 degrees), then you will likely need to use the unit circle to solve the equation. Otherwise, you can use the basic trigonometric identities.

5. Are there any tips for solving trigonometry equations more efficiently?

One tip is to always check for common factors or patterns in the equation that can be simplified or factored. Additionally, it can be helpful to draw a diagram or use a reference triangle to visualize the problem and better understand the trigonometric relationships involved.

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