Evaluate 2 tan 240 + 3 cos120 leaving your answer in a surd form
We use the fact that:
- tan(240) = tan(240 - 180) = -tan(60) = -sqrt(3)
- cos(120) = -cos(120 - 360) = -cos(240) = -1/2
So, we have:
2 tan 240 + 3 cos120 = 2(-sqrt(3)) + 3(-1/2)
= -2sqrt(3) - 3/2
= (-4sqrt(3) - 6)/2
= -2(2sqrt(3) + 3)/2
= -(2sqrt(3) + 3)
First, we need to find the values for tan(240) and cos(120):
240 degrees is in the third quadrant where both cosine and sine values are negative. To find the values, we can subtract 240 - 180 = 60. So tan(240) = -tan(60).
tan(60) is known to be √3, therefore tan(240) = - √3.
120 degrees is in the second quadrant where only sine values are positive. To find the values, we can subtract 180 - 120 = 60. So cos(120) = -cos(60).
cos(60) is known to be 1/2, therefore cos(120) = - 1/2.
Now we can plug these values back into the expression:
2 * (-√3) + 3 * (-1/2) = -2√3 - 3/2.
To evaluate the expression 2 tan 240 + 3 cos 120, we need to find the values of tan 240 and cos 120.
Step 1: Finding tan 240
We can use the fact that tangent is equal to sine divided by cosine. So, we have tan 240 = sin 240 / cos 240.
Step 2: Finding sin 240 and cos 240
To find the values of sin 240 and cos 240, we can use the unit circle or trigonometric identities. In this case, let's use the unit circle.
Looking at the unit circle, we can see that the coordinates for the point corresponding to the angle 240 degrees are (-√3/2, -1/2).
Therefore, sin 240 = -1/2 and cos 240 = -√3/2.
Step 3: Calculating tan 240
Using the values found above, we can calculate tan 240:
tan 240 = sin 240 / cos 240 = (-1/2) / (-√3/2) = 1 / √3
Step 4: Finding cos 120
To find cos 120, we can again use the unit circle or trigonometric identities.
Looking at the unit circle, the coordinates for the point corresponding to the angle 120 degrees are (-1/2, √3/2).
Therefore, cos 120 = -1/2.
Step 5: Evaluating the expression
Now we can substitute the values we found into the expression:
2 tan 240 + 3 cos 120 = 2 * (1 / √3) + 3 * (-1/2) = 2/√3 - 3/2
Thus, the expression 2 tan 240 + 3 cos 120 evaluates to (2/√3) - (3/2) in surd form.
To evaluate the given expression, we will use the trigonometric identities for tangent and cosine of supplementary angles:
1. Recall that the tangent function can be written as a ratio of sine and cosine: tan(x) = sin(x) / cos(x).
2. Then, we can rewrite the expression as: 2 * (sin(240) / cos(240)) + 3 * cos(120).
3. To simplify further, let's determine the values of sin(240), cos(240), and cos(120):
- sin(240) = -sin(60) = -√3 / 2
- cos(240) = -cos(60) = -1 / 2
- cos(120) = -cos(60) = -1 / 2
4. Plug the values into our expression:
= 2 * (-√3 / 2 / -1 / 2) + 3 * (-1 / 2)
= -√3 + (-3 / 2)
= -√3 - 3 / 2
So, the evaluated expression 2 tan 240 + 3 cos120 is -√3 - 3 / 2 in surd form.