Find all angles θ between 0° and 180° satisfying the given equation. Round your answer to one decimal place. (Enter your answers as a comma-separated list.) cos(θ) = 7/8

Question

How do I find more than one angle when I'm using my calculator for this problem??

Answer 1

This problem only has one solution within 0°≤ Θ ≤ 180°

specifically, cosΘ =7/8 → Θ = cos^-1(7/8) ≈ 29.0º

However, if on the other hand your problem was something like sinΘ = 7/8, then you'd have two solutions for Θ.

Ex: sinΘ = 7/8

Θ = sin^-1(7/8) ≈ 61.0°

Now, since sinΘ is positive in both Q1 and Q2, then the second answer would be in Q2. This angle would also make have a 61.0º reference angle. Meaning that the angle that your looking for in Q2 is 180º - 61.0º = 119.0º

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In your case, cosΘ is only positive in Q1 and Q4. So if your restrictions were 0º ≤ Θ < 360°, then your second answer for cosΘ = 7/8 would be 360º - 29.0º = 331.0º

Basically, if you are trying to find an angle in

Q1 your answer is Θ

Q2 your answer is 180° - Θ

Q3 your answer is 180° + Θ

Q4 your answer is 360º - Θ

If you want more examples look at your unit circle values:

45º, 135º, 225º, 315º all follow this pattern

30º, 150º, 210º, 330º

and so on...they all follow the pattern above.

Hope this helps.

Mr. K

Answer 2

If your calculator has a key for cos^-1, this may be a second level function on your calculator

First for ease of data entry you could probably change your fraction to a decimal.

That decimal is 0.875

cos^-1(0.875) = will give you an acute angle in degrees whose cosine is 0.875

cos^-1(0.875) = 28.95502437°

Keep in mind the Quadrants when you are looking for angles between 0 and 180 degrees. A positive cosine in this range is in quadrant I. Angles greater than 90 but less than 180 in quadrant II will have a negative cosine. (180 - 28.95502437) has a negative 7/8 as its cosine.

Since you will be rounding your answers, you could try a couple of values around the angle above with the cosine function on your calculator.

Give it a try

Answer 3

You can't use your calculator for these type problems. Your calculator gives only one of the answers. To do these problems, you need to be familiar with the unit circle.

Since cos(θ) = adjacent/hypotenuse we can assume the adjacent side is 7 and the hypotenuse is 8. The hypotenuse is always positive so a cosine of 7/8 (a positive value) means the 7 must be positive. There are two locations on the circle where this can occur:

However, the lower one (θ₂) does not meet the requirements of the domain (angles between 0° and 180°) so, in this case, there is only one answer. Using your calculator, you get 29.0° when rounded to one decimal which is θ₁ in my drawing

Answer 4

cosx=7/8 cosine is an even function so cosx = cos(-x) = cos(-x+2pi) -x+2p < 2pi = 180 degrees

x=cos^-1(7/8) = 0.5053 = .5 pi radians 1 radian = 57.3 degrees 1/2 radian=57.3/2 = 28.65 degrees

converting to degrees x=(-.5 radian x 180)/pi =-1/2 x180/pi = (1/2)57.3 = 28.7 degrees, rounded off to one decimal

Then the other angle 0< -x+2pi < 2pi=180 is -28.7+180= 151.3 degrees

Find the easy angle with your calculator, 28.7, then just subtract that from 180 degrees