Find cot θ if csc θ =√17/4 and tan θ > 0?
1. Since “csc” is defined as “hypotenuse over opposite,”
draw a right triangle where sqrt(17) is the hypotenuse and
4 is the side opposite theta. Then, since “cot” is defined as
“adjacent over opposite,” you need to find the missing side.
Using the Pythagorean theorem, the missing side is
sqrt(17 – 16) = 1. So cot(theta) = 1/4.
#2. Since “cot” is adj/opp and “sin” is opp/hyp, their product is
adj/hyp, which is the cosine. Also sin(pi/2 – x) = cos(x). So
the whole expression is
cos x – cos x + cos x = cos x.
#3. “sec” = hyp/adj, so draw a right triangle where sqrt(37) is the
hypotenuse and 6 is the side adjacent to theta. Since “tan” is opp/adj,
you need to find the missing side using the Pythagorean Theorem:
the side opposite theta is sqrt(37 – 36) = 1. So tan(theta) could be
1/6, but the fact that sin(theta) < 0 means the angle is in the 3rd or 4th
quadrant. It can’t be in the third quadrant, because the secant was > 0.
So it’s in the 4th quadrant, and tan(theta) = -1/6.
#4. sec^2(x) – 2 = tan^2(x)
You have a Pythagorean identity that says sec^2(x) – 1 = tan^2(x).
So the left side can be rewritten as tan^2(x) – 1.
Which means the equation will say -1 = 0,
and it has no solutions.
#5. In a 30-60-90 triangle, the short side is 1/2 of the hypotenuse,
and the leg perpendicular to the short side is sqrt(3)/2 of the hypotenuse.
Another way to look at it is to draw a 30-60-90 triangle and label the
hypotenuse “2”, the shortest leg “1”, and the other leg “sqrt(3)”.
The sqrt(3) is opposite the 60 degree angle, and the 2 is the hypotenuse,
so sin(60) = sqrt(3)/2. In radian measure, sin(pi/3) = sqrt(3)/2.
The sine is positive in the 2nd quadrant as well, so another solution is
sin(2 pi/3) = sqrt(3)/2. These two angles (pi/3 and 2pi/3) are the only
angles in the interval (0,2 pi) that have sqrt(3)/2 as their sine.
#6. You have an angle-sum identity that says
sin (A-B) = sin A cos B – cos A sin B.
Evidently the expression in this problem is sin (9x – x),
so it’s sin(8x).
#7. Another angle-sum identity says
cos(A-B) = cos A cos B + sin A sin B,
so the expression given in #7 is cos(112-45) = cos(67 degrees)
#8. sin 2x = sin (x+x) = sin x cos x + cos x sin x
cos 2x = cos (x+x) = cos x cos x – sin x sin x
So, sin 2x – cos 2x = sin x cos x + cos x sin x + cos x cos x – sin x sin x.
It’s not obvious from the question whether someone wants you to write this as
2 sin x cos x + cos^2 x – sin^2 x.
#9. Since cos(2a) = cos^2(a) – sin^2(a), and
since sin^2(a) + cos^2(a) = 1, we have
cos(2a) = 2 cos^2(a) – 1.
Now just let a = 22.5 degrees, and you have
cos(45) = 2 cos^2 (22.5) – 1
sqrt(2)/2 = 2 cos^2 (22.5) – 1
sqrt(2)/2 + 1 = 2 cos^2 (22.5)
1/2 + sqrt(2)/4 = cos^2 (22.5)
sin^2 (22.5) = 1 – [ 1/2 + sqrt(2)/4) ]
sin^2 (22.5) = 1/2 – sqrt(2)/4
sin (22.5) = sqrt(1/2 – sqrt(2)/4)
This is the exact value; what they DON’T want is a decimal value,
which will necessarily be rounded off.
#10. Note that secx = 1/cosx, so you have
(cosx)/(1+sinx) + (1+sinx)/(cosx) = 2/(cosx)
Multiply all three terms by the common denominator,
which is (cosx) (1+sinx). That gives you
cos^2(x) + (1+sin x)^2 = 2 (1+ sin x)
cos^2(x) + 1 + 2 sin x + sin^2(x) = 2 + 2 sin x
Recalling the Pythagorean identity “cos^2(x) + sin^2(x) = 1,” you have
1 + 1 + 2 sin x = 2 + 2 sin x
which is obviously true.
If csc theta=4/3, what is the sin, cos, tan, sec, and cot? | Socratic – Since #csc theta = 1/sintheta = "hypotenuse"/"opposite"=c/a = 4/3#, this means that #a# and #c# are multiples of #3# and #4#, respectively. In other words, we have #c=4k# and #a=3k#, for a real number #k#.Use the definition of tangent to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values. Since the opposite and adjacent sides are known, use the Pythagorean theorem to find the remaining side. Replace the known values in the equation.I am graphing sin, cos, csc,sec, cot, tan. I don't know how to do the cot, or tan. I need someone to explain those. If #csc theta=4/3#, what is the sin, cos, tan, sec, and cot?
Find Trig Functions Using Identities tan(theta)=-3/4 , cos… | Mathway – Then, since "cot" is defined as. "adjacent over opposite," you need to find the missing side. Using the Pythagorean theorem, the missing side is. So the left side can be rewritten as tan^2(x) – 1. Which means the equation will say -1 = 0, and it has no solutions. #5. In a 30-60-90 triangle, the short side is…In this lesson we are going to learn how to graph the other four trigonometric functions: tan, cot, sec, and csc. What is most interesting about all four of these graphs is that we encounter discontinuity!Find the values of the six trigonometric functions of θ. (If an answer is undefined, enter UNDEFINED.) Function Value: csc(θ) = 4 Constraint: cot(θ) < 0 What is a simplified form of the expression [sec^2x-1]/[(sinx)(secx)]? a. cot x b. csc x c. tan x***** d. sec x tan x I think this is the correct answer, but I do…
How do you find sin, cos, tan, sec, csc, and cot given… – HomeworkLib – These are found by taking ratios between the same sides shown above, except reversing the Explain why the csc of an angle will always be greater than 1. Use your knowledge of 45-45-90 triangles Use your knowledge of 30-60-90 triangles to find the cosecant, secant, and cotangent of a 30 degree angle.Cotangent theta is the adjacent over opposite. We use the pythagorean theorem to find for the adjacent side. a = √17 – 4^2 a = 1.\cot. \csc. \sec. \alpha. \times. \arctan. \tan.