Find Solutions to Tan x = Cos x in Radians

[Mark53]

Homework Statement

Find all numbers x ∈ [0, 2π] satisfying tan x = cos x. Your answers should be expressed in radians, rounded to 4 decimal places. Show all your working.

[You will need to use a scientific calculator that has buttons such as sin−1 or arcsin so as to be able to find the angles for which the sin function attains given values.

The Attempt at a Solution

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I don't know where to start I have had a look at the unit circle but can't see anywhere where they are equal

 

[Chestermiller]

The hint they gives you suggests that you should be solving your equation for sin x. What is tan x in terms of sin x and cos x? If you substitute for tan x in your equation, what do you get?

 

[Charles Link]

Suggestion: Express $\tan{x}=\sin{x}/\cos{x}$. Also $\$ $\cos^2{x}=1-\sin^2{x}$. Much simpler to work with the $sin{x}$ than to graph $y= \tan{x}$ and $y= \cos{x}$. A graphical solution might be a good check for the solution.

 

[Mark53]

The hint they gives you suggests that you should be solving your equation for sin x. What is tan x in terms of sin x and cos x? If you substitute for tan x in your equation, what do you get?

tan x = sinx/cosx

which means

sinx/cosx=cosx

sinx=cos^2x

Is this what you mean?

 

[Chestermiller]

tan x = sinx/cosx

which means

sinx/cosx=cosx

sinx=cos^2x

Is this what you mean?

Yes. Now, what is cos^2 x in terms of sin^2x?

 

[Mark53]

Yes. Now, what is cos^2 x in terms of sin^2x?

that means

sinx=1-sin^2x

 

[Chestermiller]

that means

sinx=1-sin^2x

Good. Now solve this quadratic equation for sin x using the quadratic formula.

 

[SammyS]

tan x = sinx/cosx

which means

sinx/cosx=cosx

sinx=cos^2x

Is this what you mean?

It's clearer to write cos^2(x). Even better to use the Superscript feature, X², to give cos² x.

 

[Mark53]

Good. Now solve this quadratic equation for sin x using the quadratic formula.

does that mean

sin x = (-1 +sqrt5)/2 or (1 +sqrt5)/2

 

[Chestermiller]

does that mean

sin x = (-1 +sqrt5)/2 or (1 +sqrt5)/2

One of these roots is >1. As a decimal, what is the other root. What angles does your calculator say that this corresponds to on the interval between x = 0 and x = 2pi?

 

[Mark53]

One of these roots is >1. As a decimal, what is the other root. What angles does your calculator say that this corresponds to on the interval between x = 0 and x = 2pi?

0.618 which means arcsin 0.618 = 0.6662

is this my answer?

 

[Charles Link]

0.618 which means arcsin 0.618 = 0.6662

is this my answer?

If I may offer a hint here=.666 radians is about 40 degrees. Is that the only place where sin(x)=.618?

 

[Chestermiller]

0.618 which means arcsin 0.618 = 0.6662

is this my answer?

Not good enough. There is another angle on the interval that has this same value of sine.

 

[Mark53]

Not good enough. There is another angle on the interval that has this same value of sine.

it would be in the second quadrant but how do i calculate it

 

[Chestermiller]

it would be in the second quadrant but how do i calculate it

Using the sine of the difference between two angles formula, what is $\sin(\pi -\theta)$?

 

[Mark53]

Using the sine of the difference between two angles formula, what is $\sin(\pi -\theta)$?

does this mean that the other answer would be 3.8078?

 

[Mark44]

does this mean that the other answer would be 3.8078?

No. It looks like you added your first answer to $\pi$, rather than subtracting it from $\pi$.

 

[Mark53]

No. It looks like you added your first answer to $\pi$, rather than subtracting it from $\pi$.

so the answer would be 2.48 radians

 

[Mark44]

so the answer would be 2.48 radians

You can check both your answers by substituting them in your equation: $\tan(x) = \cos(x)$. The left and right sides should be equal for those two numbers. Your calculator should be in radian mode, though.

 

[Mark44]

Since you have rounded your answers (or at least haven't written all the digits shown on your calculator), the left and right sides of the equation will only be close, not exactly the same.

 

[Mark53]

Since you have rounded your answers (or at least haven't written all the digits shown on your calculator), the left and right sides of the equation will only be close, not exactly the same.

Thanks for the help