# How Can These Trigonometric Equations Be Solved?

• MHB
• matyw
In summary: Using a calculator I get $\cos^{-1}\left(-\dfrac47\right)\approx 107.6^\circ$ and $\cos^{-1}\left(\dfrac12\right)=60^\circ$.In summary, we solved three trigonometry equations. For the first, we used the identities $\sin(A+B)=\sin(A)\cos(B)+\sin(B)\cos(A)$ and $\sin(B)=\pm\sqrt{1-\cos^2(B)},\quad\cos(A)=\pm\sqrt{1-\sin^2(A)}$. For the second, we rewrote the equation in terms of $\sin(x)$ and solved for the root $\sin(x)=1$. For the
matyw
Hi all

1. find the value of sin(A+B) when sin A = 1/2 and cos B = 3/8

2. solve the equation: 2 - 2 sinx = cos²X when -180° < X < 180°

3. solve rquation for X: 2 tan°X + 2 sec°X - sec X = 12

any help would be great
Cheers
Mat

Hello matyw and welcome to MHB! :D

As these are trigonometry questions I've moved your thread into the Trigonometry forum.

Also, we ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?

greg1313 said:
Hello matyw and welcome to MHB! :D

As these are trigonometry questions I've moved your thread into the Trigonometry forum.

Also, we ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?

Hi Greg

I actually don't know where to start hence why i have posted the equations in full.
My thoughts were to see the answers in full working order, so i could work backwards to see the processes people take to come up with a result. It is hard to be teaching yourself especially when you have no idea of what the results are meant to be
My apologies, i do understand if i don't get help with this.

Regards
Mat

These questions assume some background knowledge of trigonometry, specifically (for question 1)

$$\sin(A+B)=\sin(A)\cos(B)+\sin(B)\cos(A)$$

and

$$\sin(B)=\pm\sqrt{1-\cos^2(B)},\quad\cos(A)=\pm\sqrt{1-\sin^2(A)}$$

Working with angles in the first quadrant (do you know what the first quadrant is?), can you now solve question 1?

It's somewhat difficult to post good help without knowing what trigonometry you are familiar with. Can you post a brief summary of your trigonometry knowledge?

It's been a while and the OP has not responded so I'll post some work.

matyw said:
1. Find the value of sin(A+B) when sin A = 1/2 and cos B = 3/8.

Assuming $A$ and $B$ are in quadrant I,

$$\sin(B)=\sqrt{1-\dfrac{9}{64}}=\dfrac{\sqrt{55}}{8}\quad\cos(A)=\sqrt{1-\dfrac14}=\dfrac{\sqrt3}{2}$$

$$\sin(A+B)=\sin(A)\cos(B)+\sin(B)\cos(A)$$

$$=\dfrac12\cdot\dfrac38+\dfrac{\sqrt{55}}{8}\cdot\dfrac{\sqrt3}{2}=\dfrac{3}{16}+\dfrac{\sqrt{165}}{16}=\dfrac{1}{16}\left(3+\sqrt{165}\right)$$

matyw said:
2. Solve the equation 2 - 2 sinx = cos²X when -180° < X < 180°

$$2-2\sin(x)=\cos^2(x)$$

$$2-2\sin(x)=1-\sin^2(x)$$

$$\sin^2(x)-2\sin(x)+1=0$$

$$(\sin(x)-1)^2=0$$

$$\Rightarrow\sin(x)=1,\quad x=90^\circ$$

matyw said:
3. solve rquation for X: 2 tan°X + 2 sec°X - sec X = 12

I'm going to assume this is $2\tan^2(x)+2\sec^2(x)-\sec(x)=12$:

$$2\tan^2(x)+2\sec^2(x)-\sec(x)=12$$

$$2(\sec^2(x)-1)+2\sec^2(x)-\sec(x)-12=0$$

$$4\sec^2(x)-\sec(x)-14=0$$

$$(4\sec(x)+7)(\sec(x)-2)=0$$

$$x=\cos^{-1}\left(-\dfrac47\right),\quad x=\cos^{-1}\left(\dfrac12\right)$$

## 1. How do you solve an equation?

To solve an equation, you must isolate the variable on one side of the equal sign. You can do this by using inverse operations, such as addition, subtraction, multiplication, and division, to cancel out any constants or coefficients attached to the variable. Once the variable is isolated, the solution can be easily determined.

## 2. What is the order of operations for solving equations?

The order of operations for solving equations is similar to the order of operations for any math problem: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). However, when solving equations, the goal is to isolate the variable, so you may need to use inverse operations to rearrange the equation.

## 3. Can you solve an equation without using inverse operations?

No, inverse operations are crucial for solving equations. They allow you to isolate the variable and find the solution. Without using inverse operations, the equation will not be solved correctly.

## 4. Are there any shortcuts for solving equations?

Yes, there are a few shortcuts that can sometimes be used when solving equations. For example, if the equation is already in the form of "variable = number," then the solution is simply the number. Additionally, if the equation contains fractions, you can multiply both sides by the least common denominator to eliminate the fractions.

## 5. How do you know if your solution is correct?

To check if your solution is correct, you can plug it back into the original equation and see if it makes the equation true. If it does, then your solution is correct. You can also use a calculator to evaluate both sides of the equation and see if they are equal.

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