Part 1)
Evaluate tan 7.5
Part 2)
Evaluate exactly cos (-(45pi(4)))
1. I recommend usng a calculator (in radians mode) or search Google with
< tan 7.5 = > in the search window.
2. Remember that you can subtract multiples of 2 pi from the argument of any trig function and get the same answer.
-45 pi*4 = -180 pi; the cosine of that is the same as the cosine of zero.
Part 1:
To evaluate tan 7.5, you can use a scientific calculator or computer software that has trigonometric function capabilities. Most calculators and computers have a built-in function for tangent.
If you don't have access to a calculator or computer software, you can also use trigonometric identities and approximations to estimate the value of tan 7.5. One such method is to convert 7.5 degrees to radians and then use the Taylor series expansion of the tangent function.
First, convert 7.5 degrees to radians. Since π radians is equivalent to 180 degrees, we can use the formula:
angle_in_radians = angle_in_degrees * (π/180)
7.5 degrees = 7.5 * (π/180) radians
Next, we can use the Taylor series expansion of the tangent function to approximate its value:
tan(x) = x + (x^3)/3 + (2 * x^5)/15 + (17 * x^7)/315 + ...
In this case, x represents the angle in radians, which is 7.5 * (π/180).
By plugging in the value of x into the series and adding up a reasonable number of terms, you can approximate the value of tan 7.5.
Part 2:
To evaluate exactly cos (-(45π(4))), we can simplify the expression step by step.
First, let's simplify -(45π(4)):
-(45π(4)) = -45π
Next, we'll simplify cos(-45π) by using the properties of the cosine function. The cosine function has a period of 2π, meaning that the value of cos(x) repeats every 2π.
Since -45π represents an angle that is 45π radians counter-clockwise from the positive x-axis, we can use the periodicity of the cosine function to simplify the expression. The cosine value at -45π will be the same as the cosine value at -45π + 2π, which is equivalent to -45π + 6π.
-45π + 6π = -39π
Therefore, cos (-(45π(4))) is equivalent to cos(-39π).
Now, evaluating cos(-39π) exactly depends on the context or further instructions provided. If you are expected to provide a numerical approximation, you can use a calculator or computer software to find the cosine of -39π radians. However, if you need to express the result as a simplified exact value, you can use trigonometric identities and manipulation to simplify the expression further.