cos[(tan^-1(3/4))+(cos^-1(9/41))]
To solve this expression, let's break it down step by step:
Step 1: Start with the innermost trigonometric functions.
The given expression is:
cos[(tan^-1(3/4)) + (cos^-1(9/41))]
Let's simplify the inner functions one by one.
Inner function 1: tan^-1(3/4)
To find the value of this expression, take the inverse tangent (arctan) of 3/4:
tan^-1(3/4) ≈ 0.6435
Inner function 2: cos^-1(9/41)
To find the value of this expression, take the inverse cosine (arccos) of 9/41:
cos^-1(9/41) ≈ 1.3239
Step 2: Substitute the values obtained from the inner functions into the original expression.
cos[(tan^-1(3/4)) + (cos^-1(9/41))]
≈ cos(0.6435 + 1.3239)
Step 3: Combine the angles inside the cosine function.
Since cos(a + b) = cos(a) * cos(b) - sin(a) * sin(b), we can rewrite the expression as follows:
cos(0.6435 + 1.3239)
= cos(0.6435) * cos(1.3239) - sin(0.6435)*sin(1.3239)
Step 4: Evaluate the trigonometric functions using a calculator.
Using a calculator, we find:
cos(0.6435) ≈ 0.7902
cos(1.3239) ≈ 0.2358
sin(0.6435) ≈ 0.6124
sin(1.3239) ≈ 0.7900
Step 5: Substitute the calculated values back into the expression.
cos(0.6435 + 1.3239)
≈ (0.7902) * (0.2358) - (0.6124) * (0.7900)
≈ 0.1859 - 0.4837
≈ -0.2978
Therefore, the value of cos[(tan^-1(3/4)) + (cos^-1(9/41))] is approximately -0.2978.
To find the value of cos[(tan^-1(3/4)) + (cos^-1(9/41))], we can follow these steps:
Step 1: Let's first find the values of tan^-1(3/4) and cos^-1(9/41).
- The expression tan^-1(3/4) represents the angle whose tangent is 3/4.
- Similarly, the expression cos^-1(9/41) represents the angle whose cosine is 9/41.
Step 2: Use a scientific calculator or trigonometric identity tables to evaluate the inverse trigonometric functions.
- tan^-1(3/4) is approximately equal to 37.04 degrees (or approximately 0.6435 radians).
- cos^-1(9/41) is approximately equal to 67.03 degrees (or approximately 1.167 radians).
Step 3: Now, we substitute the values back into the original expression: cos[(tan^-1(3/4)) + (cos^-1(9/41))].
- cos[(tan^-1(3/4)) + (cos^-1(9/41))] = cos[(37.04°) + (67.03°)] (or cos[(0.6435) + (1.167)] in radians).
Step 4: Add the angles: 37.04° + 67.03° = 104.07° (or 1.8105 radians).
Step 5: Evaluate the cosine of the sum of the angles: cos(104.07°) (or cos(1.8105) in radians).
- Use a scientific calculator or trigonometric identity tables to find cos(104.07°) (or cos(1.8105) in radians).
- The cosine value is approximately 0.453 (or rounded to three decimal places).
Therefore, cos[(tan^-1(3/4)) + (cos^-1(9/41))] is approximately equal to 0.453 (rounded to three decimal places).
let tan^-1 (3/4) = A and let cos^-1 (9/41) = B
then cos[(tan^-1(3/4))+(cos^-1(9/41))]
= cos(A + B)
= cosA cosB - sinA sinB
IF tan^-1 (3/4) = A
then tanA = 3/4
---> sinA = 3/5 and cosA = 4/5, recognize the 3-4-5 right angled triangle
if cos^-1 (9/41) = B
then cosB = 9/41
---> x^2 + y^2 = r^2
81 + y^2 = 1681
y^2 = 1600
y = √1600 = 40 and thus sinB = 40/41
so back to
cosA cosB - sinA sinB
= (4/5)(9/41) - (3/5)(40/41) = -84/205
( I tested my answer with my calculator, it is correct)