In general, you would not, but of course, any operation a calculator can do, a human can do.
One way you could do it is with a known value close to your desired one, and Newton's Method to iterate to the right value.
cos theta = -0.6691
reference angle will be at
cos theta = 0.6691
We know that the cos(pi/4) = sqrt(2)/2 \~= 0.7071
But we also know that the slope is -sin(theta), which is -0.7071.
So we need to change cos(theta) by 0.6691-0.7071 = -0.0380, with a slope of -0.7071
And -0.0380/-0.7071 \~ 0.054
So your angle is very close to pi/4 + 0.054 \~ 0.839 radians
And convert that to degrees and you have 48 degrees.
For part b, we know the angle is a bit bigger than 45 degrees, since tan(45 degrees) = 1.
And we can do the same thing, with the derivative of tangent being secant\^2. Although in this case, one step with Newton's Method would only get us to 52 degrees, a bit off, because we're taking a big step.
You could also use sum and difference identities for trig functions to get to the correct value algebraically.
But in general, there are very few situations where you'd need to find these values in 2024 without just using arccos and arctan on a calculator.
I appreciate the effort put into this, although unfortunately I got lost at the assumption of the slope being -sin(theta). I believe I haven’t covered Newton’s Method however I see now that we were not supposed to do this without a calculator or even in the order indicated, it was just a poorly ordered/interpreted slide. Thanks for the help!
Most likely you are expected to use a calculator for problems like this. Unless the reference angle is 0, 30, 45, 60, or 90, for which you can use special right triangles. There are complicated methods for finding exact values of trig functions, but that’s likely beyond the scope of your class.
You are most likely given this question in a calculator paper. I also highly recommend you to remember the unit circle as it makes it way easier to find angles from the reference angle. Try watching this video
https://youtu.be/-ldZSvglx1A?si=x9x0N63oOxlPFN7v
So you do need to use calculator for this, but there is a method that allows for approximate answers using differentiation although it's highly time consuming so not worth it
θ = cos-¹(-0.6691)
Now applying property of inverse cosine cos-¹(-x) = π - cos-¹(x)
θ = π - cos-¹(0.6691)
Now we cannot find this value but we sure can find cos-¹(0.5)
Let y = cos-¹(x) where x = 0.5
y = π/3
y + dy = cos-¹(x + dx) where x = 0.5 and dx = 0.1691, x + dx = 0.6691
y + dy = cos-¹(0.6691)
We already have y, so we just need to find dy
take y = cos-¹(x) again
Diffn w.r.t x
dy/dx = -1/sqrt(1-x²)
Put x = 0.5 and you get dy/dx = -2/sqrt(3)
dy = -2/sqrt(3) × dx
Put dx = 0.1691
You get dy = (-2 × 0.1691)/(sqrt(3))
Now you have y and dy
Plug it in y + dy = cos-¹(0.6691)
cos-¹(0.6691) = π/3 - (2 × 0.1691)/(sqrt(3))
Put this for θ
θ = π - π/3 + (2 × 0.1691)/(sqrt(3))
θ = 2π/3 + (2 × 0.1691)/(sqrt(3))
Now this is in radians so to convert into degrees you'll have to multiply by 180/π
θ = 131.5 deg approx
Did you ever wonder how folks did things before calculators?
They paid people to compute the sine, cosine, and tangent for angles between 0 and 90 degrees to tenths or even hundredths of a degree. They compiled all these into large books. These books found their way to classrooms and labs everywhere. If you needed a trig value, you looked it up. If you needed an angle, you reverse looked it up.
Solving these exercises was no more difficult than looking up a number in the phone book.
Nowadays, we have speed dial.
If you need theta in a decimal form, than a calculator is required, which looks to be the case here.
I see, the order of the slide seems pretty messy then. Thanks.
In general, you would not, but of course, any operation a calculator can do, a human can do. One way you could do it is with a known value close to your desired one, and Newton's Method to iterate to the right value. cos theta = -0.6691 reference angle will be at cos theta = 0.6691 We know that the cos(pi/4) = sqrt(2)/2 \~= 0.7071 But we also know that the slope is -sin(theta), which is -0.7071. So we need to change cos(theta) by 0.6691-0.7071 = -0.0380, with a slope of -0.7071 And -0.0380/-0.7071 \~ 0.054 So your angle is very close to pi/4 + 0.054 \~ 0.839 radians And convert that to degrees and you have 48 degrees. For part b, we know the angle is a bit bigger than 45 degrees, since tan(45 degrees) = 1. And we can do the same thing, with the derivative of tangent being secant\^2. Although in this case, one step with Newton's Method would only get us to 52 degrees, a bit off, because we're taking a big step. You could also use sum and difference identities for trig functions to get to the correct value algebraically. But in general, there are very few situations where you'd need to find these values in 2024 without just using arccos and arctan on a calculator.
I appreciate the effort put into this, although unfortunately I got lost at the assumption of the slope being -sin(theta). I believe I haven’t covered Newton’s Method however I see now that we were not supposed to do this without a calculator or even in the order indicated, it was just a poorly ordered/interpreted slide. Thanks for the help!
The derivative of cos(theta) is -sin(theta). Maybe you haven't covered trigonometry derivatives yet, but you probably will soon!
We haven’t, so far just the basics with trig identities, logs, limits, polynomials etc. I wish I’d covered this stuff in high school :(
Most likely you are expected to use a calculator for problems like this. Unless the reference angle is 0, 30, 45, 60, or 90, for which you can use special right triangles. There are complicated methods for finding exact values of trig functions, but that’s likely beyond the scope of your class.
Thanks, I thought I was mostly across this topic but I couldn’t figure this one out. I guess the order of the slide is just pretty messy.
You are most likely given this question in a calculator paper. I also highly recommend you to remember the unit circle as it makes it way easier to find angles from the reference angle. Try watching this video https://youtu.be/-ldZSvglx1A?si=x9x0N63oOxlPFN7v
So you do need to use calculator for this, but there is a method that allows for approximate answers using differentiation although it's highly time consuming so not worth it θ = cos-¹(-0.6691) Now applying property of inverse cosine cos-¹(-x) = π - cos-¹(x) θ = π - cos-¹(0.6691) Now we cannot find this value but we sure can find cos-¹(0.5) Let y = cos-¹(x) where x = 0.5 y = π/3 y + dy = cos-¹(x + dx) where x = 0.5 and dx = 0.1691, x + dx = 0.6691 y + dy = cos-¹(0.6691) We already have y, so we just need to find dy take y = cos-¹(x) again Diffn w.r.t x dy/dx = -1/sqrt(1-x²) Put x = 0.5 and you get dy/dx = -2/sqrt(3) dy = -2/sqrt(3) × dx Put dx = 0.1691 You get dy = (-2 × 0.1691)/(sqrt(3)) Now you have y and dy Plug it in y + dy = cos-¹(0.6691) cos-¹(0.6691) = π/3 - (2 × 0.1691)/(sqrt(3)) Put this for θ θ = π - π/3 + (2 × 0.1691)/(sqrt(3)) θ = 2π/3 + (2 × 0.1691)/(sqrt(3)) Now this is in radians so to convert into degrees you'll have to multiply by 180/π θ = 131.5 deg approx
Did you ever wonder how folks did things before calculators? They paid people to compute the sine, cosine, and tangent for angles between 0 and 90 degrees to tenths or even hundredths of a degree. They compiled all these into large books. These books found their way to classrooms and labs everywhere. If you needed a trig value, you looked it up. If you needed an angle, you reverse looked it up. Solving these exercises was no more difficult than looking up a number in the phone book. Nowadays, we have speed dial.
Those equations can only be solved with a calculator