Answer :

mhanifa

a) - 1/2;  c) [tex]\( \frac{2\sqrt{3}}{3} \)[/tex]; e) undefined; g) 0; i) - 1

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a) [tex]\( \cos(240^\circ) \)[/tex]

240° lies in the third quadrant of the unit circle where the cosine values are negative. Use the reference angle. The reference angle is [tex]\( 240^\circ - 180^\circ = 60^\circ \).[/tex]

Since the cosine of 60° is 1/2, and we're in the third quadrant where cosine is negative, the value of [tex]\( \cos(240^\circ) \)[/tex] is -1/2.

c) [tex]\( \csc\left(\frac{2\pi}{3}\right) \)[/tex]

[tex]\( \frac{2\pi}{3} \)[/tex] radians lies in the second quadrant where the cosecant (and thus the sine) values are positive. The reference angle for [tex]\( \frac{2\pi}{3} \)[/tex] is [tex]\( \pi - \frac{2\pi}{3} = \frac{\pi}{3} \)[/tex].

The sine of  [tex]\( \frac{\pi}{3} \)[/tex] is [tex]\( \frac{\sqrt{3}}{2} \)[/tex], and the cosecant is the reciprocal of the sine, so the value of [tex]\( \csc\left(\frac{2\pi}{3}\right) \)[/tex] is [tex]\( \frac{2}{\sqrt{3}} \)[/tex], which can be rationalized to [tex]\( \frac{2\sqrt{3}}{3} \)[/tex].

e) [tex]\( \tan(-90^\circ) \)[/tex]

The tangent function is undefined at [tex]\( -90^\circ \)[/tex] (and [tex]\( 90^\circ \)[/tex]) because it involves dividing by zero (since [tex]\( \cos(90^\circ) = 0 \)[/tex]), which is not possible. Therefore, the value is "undefined".

g) [tex]\( \cos\left(\frac{17\pi}{2}\right) \)[/tex]

To find the cosine, we can simplify the angle by subtracting multiples of [tex]\( 2\pi \)[/tex] since cosine is periodic with a period of [tex]\( 2\pi \)[/tex].

[tex]\( \frac{17\pi}{2} \)[/tex] is equivalent to [tex]\( 8\pi + \frac{\pi}{2} \)[/tex], which is  4 full rotations plus [tex]\( \frac{\pi}{2} \)[/tex]. The cosine of [tex]\( \frac{\pi}{2} \)[/tex] is 0, so the value of [tex]\( \cos\left(\frac{17\pi}{2}\right) \)[/tex] is also 0.

i) [tex]\( \cot\left(-\frac{5\pi}{4}\right) \)[/tex]

Since cotangent is the reciprocal of tangent, we can first find the tangent of the angle and then take the reciprocal.

[tex]\( -\frac{5\pi}{4} \)[/tex] is equivalent to [tex]\( -\pi - \frac{\pi}{4} \)[/tex], or one full rotation minus [tex]\( \frac{\pi}{4} \)[/tex].  Since a full rotation doesn't change the value of a trigonometric function, we can simply evaluate [tex]\( \tan\left(\frac{\pi}{4}\right) \)[/tex] and take the negative value (since we started with a negative angle).

The tangent of [tex]\( \frac{\pi}{4} \)[/tex] is 1, so [tex]\( \tan\left(-\frac{\pi}{4}\right)[/tex] is -1, and therefore, [tex]\( \cot\left(-\frac{5\pi}{4}\right) = -1 \)[/tex] as the cotangent is the reciprocal of the tangent.

sqdancefan

Answer:

 a) cos(240°) = -1/2
  c) csc(2π/3) = 2/√3
  e) tan(-90°) = undefined
  g) cos(17π/2) = 0
  i) cot(-5π/4) = -1

Step-by-step explanation:

You want the values of the trig functions listed.

Diagram

The given diagram can be filled in with angles and values as in the attached.

Trig relations

(a) The value for cos(240°) can be read from the diagram. The coordinates of a point on the units circle are (cos, sin), so the cosine of 240° is the x-coordinate of the point at that angle.

(c) The cosecant function is the reciprocal of the sine function, so the value of csc(2π/3) is found by inverting the y-coordinate of the point at that angle.

(e) The tangent function is undefined for odd multiples of 90°.

(g) The cosine is zero for odd multiples of π/2, so will be zero at 17π/2.

(i) The positive angle corresponding to -5π/4 is (-5π/4 +2π) = 3π/4. The cotangent is the ratio of cosine to sine. At this point, both have the same magnitude, but different signs, so the ratio is -1.

 a) cos(240°) = -1/2
  c) csc(2π/3) = 2/√3
  e) tan(-90°) = undefined
  g) cos(17π/2) = 0
  i) cot(-5π/4) = -1

View image sqdancefan

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