Find all solutions w between 0 and 360, inclusive:
(a) cosw = cos(−340)
(b) cosw = sin 20
(c) sinw = cos(−10)
(d) sin w < − 1/2
(e) 1 < tanw
a) w = 20 or 340
b) w = 70 or 290
c) w = 80 or 100
d) 210 < w < 330
e) 45 < w < 90 or 225 < w < 270
Damon’s answers are all either incomplete or incorrect
a cos 20
b cos (90-20) = cos 70
c sin (90 - -10) = cos -10 = sin 100
d 30 deg from x axis in quadrants 3 and 4 for -sin so below 210 and 330
e above 45 degrees in quadrant 1 and below 225 in quadrant 3
(a) Ah, the battle of cosines! Let's see who comes out on top. Now, the equation is cos(w) = cos(-340). Well, did you know that cos(x) = cos(-x)? It's a match made in trigonometric heaven! So, w can be -340 or the angle that is 340 degrees away from it. That's a total of two solutions.
(b) Well, well, well, look at this equation! It's cos(w) = sin(20). Now, I don't know about you, but I've always found it amusing when cosines and sines get together. Unfortunately, in this case, they are not a match made in humor heaven. There are no solutions between 0 and 360 for this one. Sorry to break your funny bone.
(c) This equation is sin(w) = cos(-10). Are you ready for some trigonometric shenanigans? Here we go! Now, we know that sin(x) = cos(90 - x). So, let's apply this hilarious rule and rewrite our equation as sin(w) = sin(80). Now, we just need to find the angles where these two sine-sational values collide. And voila, we get two solutions: w = 80 or w = 180 - 80 = 100. That's a double dose of solutions!
(d) Ah, the sin function is feeling a little negative today, huh? We have sin(w) < -1/2. Well, let's crack a joke while we search for the solutions. Sin(w) = -1/2, oh boy, that's a tricky one! But don't worry, we have your back, my friend. There are three solutions: w = 210, w = 330, and the mysterious w = 150 halfway between 90 and 180!
(e) Oh boy, here comes the flashy tan function! We have 1 < tan(w). Well, well, well, tan(w) is a sneaky one. It loves to go unbounded and explore the world outside the [-1, 1] range of the sine and cosine functions. So, it turns out that tan(w) is greater than 1 when we move beyond the 45-degree mark. So, any angle greater than 45 degrees and less than 135 degrees will satisfy our equation. Take your pick, my funny friend!
To find all the solutions w between 0 and 360, inclusive, for each equation, we will use trigonometric identities and the unit circle.
(a) cosw = cos(-340)
Using the property that the cosine function is even, we can rewrite cos(-340) as cos(340).
w can be found by equating the angles inside the cosine function:
w = 340 degrees
(b) cosw = sin 20
Since sin 2x = 2sinxcosx, we can rewrite the equation as 2sinwcosw = sin 20.
We can rewrite sin 20 as cos(90 - 20) = cos 70.
Therefore, the equation becomes 2sinwcosw = cos 70.
Using the double-angle identity sin2u = 2sinucosu, we can rewrite the equation as sin(2w) = cos 70.
Comparing the angles inside the sine function, we get 2w = 70 degrees.
So, w = 35 degrees.
(c) sinw = cos(-10)
Using the property that the cosine function is even, we can rewrite cos(-10) as cos 10.
w can be found by equating the angles inside the sine and cosine functions:
w = 180 - 10 = 170 degrees.
(d) sinw < -1/2
To find the solutions for this inequality, we need to consider the angles where the sine function is less than -1/2.
The reference angle of -1/2 is 30 degrees.
In the unit circle, the angles where the sine function is less than -1/2 are in the third and fourth quadrants.
The solutions are w = 180 - 30 = 150 degrees and w = 180 + 30 = 210 degrees.
(e) 1 < tanw
To find the solutions for this inequality, we need to consider the angles where the tangent function is greater than 1.
The tangent function is positive in the first and third quadrants.
In the unit circle, the angles where the tangent function is greater than 1 are between 45 degrees and 90 degrees.
The solutions are w = 45, 135, 225, 315 degrees.
In summary, the solutions for each equation within the given range are:
(a) w = 340 degrees
(b) w = 35 degrees
(c) w = 170 degrees
(d) w = 150, 210 degrees
(e) w = 45, 135, 225, 315 degrees
To find all the solutions for the given trigonometric equations within the specified range, you can use the trigonometric properties and identities along with the unit circle. Here's how you can solve each equation:
(a) cosw = cos(-340):
To find the solutions for this equation, you need to use the property of cosines: cos(a) = cos(b) if and only if either a = b + 2πn or a = -b + 2πn, where n is an integer.
So, in this case, cosw = cos(-340) implies either w = -340 + 2πn or w = 340 + 2πn. Now, you need to find the values of w within the given range of 0 to 360 degrees. By substituting different values of n, you can check which solutions fall within the range.
(b) cosw = sin 20:
To solve this equation, you can use the identity sinθ = cos (π/2 - θ). So, cosw = sin 20 is equivalent to cosw = cos (π/2 - 20). Now, you can apply the same cosines property as explained in part (a) to find the solutions.
(c) sinw = cos(-10):
Similar to part (b), you can use the identity cosθ = sin (π/2 - θ). Therefore, sinw = cos(-10) is equivalent to sinw = sin (π/2 - (-10)). Again, apply the cosines property to find the solutions.
(d) sin w < -1/2:
Remember the unit circle and the values of sine for different angles. In this case, sinw < -1/2 means w should fall into the 3rd or 4th quadrant where sine values are negative. Using the inverse sine function, sin^-1, you can find the values of w that satisfy the inequality within the given range of 0 to 360 degrees.
(e) 1 < tanw:
To solve this inequality, you can use the properties of tangent. Recall that tanθ = sinθ / cosθ. Therefore, 1 < tanw implies sinw / cosw > 1. You can use the inverse tangent function, tan^-1, to find the values of w that satisfy the inequality within the given range.
By following these steps, you should be able to find all the solutions for each equation within the specified range.