# How to Solve a Trigonometry Equation Using Identities and Alternative Methods?

• chwala
In summary, the trig identities state that,##\frac {1}{\sqrt 2}##⋅ ##sin ∅##+##\frac {1}{\sqrt 2}##⋅ ##cos ∅##=##{\sqrt 3}##⋅ ##cos ∅##+##sin ∅##tan ∅=##[\frac {\sqrt 6 -1}{1-{\sqrt 2}}]##
chwala
Gold Member
Homework Statement
Solve the equation##sin (∅+45^0)=2 cos (∅-30^0)##
giving all solutions in the interval ##0^0< ∅<180^0##
Relevant Equations
Trigonometric identities
Find the Mark scheme solution here;

Now find my approach;
Using the trig. identities It follows that,
##\frac {1}{\sqrt 2}##⋅ ##sin ∅##+##\frac {1}{\sqrt 2}##⋅ ##cos ∅##=##{\sqrt 3}##⋅ ##cos ∅##+##sin ∅##
→##sin ∅##[##\frac {1}{\sqrt 2}##-##1]##=##cos ∅##[##\frac {-1}{\sqrt 2}##+##{\sqrt 3}##]
→##sin ∅##⋅[##\frac {1-{\sqrt 2}}{\sqrt 2}]##=##cos ∅##⋅[##\frac {\sqrt 6 -1}{\sqrt 2}]##
→##tan ∅##=##[\frac {\sqrt 6 -1}{1-{\sqrt 2}}]##
## ∅##=##-74.051^0##, but we want our solutions to be in the domain, ##0^0< ∅<180^0##,
therefore, ## ∅##=##-74.051^0 + 180^0##=##105.9^0##

I would definitely be interested in another approach...

Last edited:

Dr Transport said:
Which answer are you getting? The markscheme answer is attached on this post...let me know which part of my identity isn't clear. Cheers...

No worries, I also get irked when I don't get some things right...mostly due to a lack of practice...I want to dedicate 2022 to doing more hour practise in all mathematical areas ...pure, applied and stats..and even operations research...I just have to be a little more serious...

chwala said:
I would definitely be interested in another approach...
The approach you took was the most obvious one; i.e., using the sum and difference of angles identities. Any other approach would likely be longer and more obtuse.

chwala
We can also use the half- angle approach by letting ##t##=tan ##\frac {1}{2}####∅##, using the knowledge and understanding of half-angles, then it follows that,

##\frac {1-t^2}{2t}##=##[\frac {1-\sqrt 2 }{{\sqrt 6}-1}]##

##1-t^2=-0.57153t##

##t^2-0.57153t-1=0##

##t_1=1.3257## and ##t_2=-0.75426##

taking, ##1.3257##=##tan## ##\frac {1}{2}####∅##

→##\frac {1}{2}####∅##=##tan^{-1}####1.3257##

→##\frac {1}{2}####∅##=##52.972074##

→##∅##=##2×52.972074=105.9^0##

Last edited:
A calculation by hand
$$-\tan\theta=(\sqrt{6}-1)(\sqrt{2}+1) \simeq (1.414*1.732-1)(1.414+1) \simeq 3.5$$
$$\tan(\theta-\frac{\pi}{2})=-\cot \theta \simeq \frac{1}{3.5}$$
$$\theta-\frac{\pi}{2} \simeq arctan \frac{1}{3.5} \simeq \frac{1}{3.5}-\frac{1}{3*(3.5)^3}\simeq 0.278 \simeq 15.9 degree$$

Last edited:
chwala

## 1. How do I solve a trigonometry equation?

To solve a trigonometry equation, you need to use the trigonometric functions (sine, cosine, and tangent) and their inverses (arcsine, arccosine, and arctangent) to find the unknown angle or side length. You also need to use the properties and identities of trigonometric functions to simplify the equation.

## 2. What are the steps to solve a trigonometry equation?

The steps to solve a trigonometry equation are as follows:

• Identify the unknown angle or side length in the equation.
• Use trigonometric functions and their inverses to isolate the unknown quantity.
• Apply the properties and identities of trigonometric functions to simplify the equation.
• Check your solution by plugging it back into the original equation.

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

Yes, you can use a calculator to solve a trigonometry equation. Most calculators have trigonometric functions and their inverses built-in, making it easier to solve equations involving these functions. However, it is important to understand the steps and concepts behind solving a trigonometry equation rather than relying solely on a calculator.

## 4. What are the common mistakes to avoid when solving a trigonometry equation?

Some common mistakes to avoid when solving a trigonometry equation include:

• Forgetting to convert angles from degrees to radians or vice versa.
• Using the wrong trigonometric function or inverse.
• Not applying the correct properties and identities of trigonometric functions.
• Forgetting to check your solution by plugging it back into the original equation.

## 5. How can I improve my skills in solving trigonometry equations?

To improve your skills in solving trigonometry equations, you can:

• Practice solving a variety of equations involving different trigonometric functions and identities.
• Understand the concepts and principles behind trigonometric functions and their inverses.
• Review and learn from your mistakes.
• Seek help from a tutor or teacher if you are struggling with a particular concept.

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