how do you evaluate a trig function without a calculator....?

Question

exp. cos-120, and exp. tan(5pie/3)

Answer 1

You need to know the unit circle and reference angles. It's all based on 30-60-90 and 45-45-90 triangles with hypotenuse equal to 1. The reference angle is the angle your angle makes with the x-axis. -120° is in QIII and makes a 60° angle with the x-axis. So its cosine is the same as cos 60°, except it's negative since it's in QIII. 5Л/3 is in QIV and its reference angle is Л/3 (also 60°). So its tangent is the same as tan Л/3 except the tangent is negative in QIV. Therefore:

cos -120 = -cos 60 = -1/2
tan 5Л/3 = -tan Л/3 = -√3

Answer 2

There are different strategies.
(a) Solve the infinite series for the function with the given angle (in radians). If the angles aren't too big, you don't need to compute too many terms.
(b) Use the values for 45/30 and 60 degrees.
For example, Cos - 120. You will have to know if this is positive or negative (it is the latter). There is a formula for evaluating this in terms of Cos of a positive angle less than 90 degrees. (I believe this is - Cos 60. Tan (5 pi/3) is also in this category. It is in the 4th quadrant, which is minus. The tangent is negatively reflected on the x-axis, so that it is negative Tan(pi/3) or -sqrt(3).
(c) Use half or double angle formulas which employ angles you have computed otherwise.

Answer 3

-120 is 60 with respect to x-axis and is located at Quadrant III, where x is a negative value,

we know that cos 60 = 1/2

therefore cos -120 = -1/2


2. 5pie/3 is also = 300 deg ;

300 deg is at Quadrant IV where x is positive and y is negative, therefore, if we are getting tangent of an angle, the expected answer is negative..

thus,

tan(5pie/3) = tan (-60) = -sqrt(3)

Answer 4

you're not any more able to. Calculators can superb convey logs to base 10 or base e. Rearrange the equation to placed it into words you could make sure: log(4)8 = [log(10)8]/[log(10)4] sparkling up this on your calculator and then multiply it with the help of using 2 (through actual undeniable actuality that your basic equation had a 2 out the front) desire I helpes.

Answer 5

I would use the unit circle. it ranges from 0 - 2pi.

Use right triangles and the radians associated with it to get an idea of the particular evaluation of the trig. function.