Trig help

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1. Aug 7, 2016

Mark53

1. The problem statement, all variables and given/known data
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.

3. The attempt at a solution

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

2. Aug 7, 2016

Staff: Mentor

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?

3. Aug 7, 2016

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.

4. Aug 7, 2016

Mark53

tan x = sinx/cosx

which means

sinx/cosx=cosx

sinx=cos^2x

Is this what you mean?

5. Aug 7, 2016

Staff: Mentor

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

6. Aug 7, 2016

Mark53

that means

sinx=1-sin^2x

7. Aug 7, 2016

Staff: Mentor

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

8. Aug 7, 2016

SammyS

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

9. Aug 7, 2016

Mark53

does that mean

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

10. Aug 7, 2016

Staff: Mentor

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?

11. Aug 7, 2016

Mark53

0.618 which means arcsin 0.618 = 0.6662

12. Aug 7, 2016

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

13. Aug 7, 2016

Staff: Mentor

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

14. Aug 7, 2016

Mark53

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

15. Aug 7, 2016

Staff: Mentor

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

Last edited: Aug 7, 2016
16. Aug 8, 2016

Mark53

does this mean that the other answer would be 3.8078?

17. Aug 8, 2016

Staff: Mentor

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

18. Aug 8, 2016

Mark53

19. Aug 8, 2016

Staff: Mentor

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.

20. Aug 8, 2016

Staff: Mentor

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.