Find the exact value of:
Cos(inverse tan 4/3 + inverse cos 5/13)
To find the value of cos(inverse tan(4/3) + inverse cos(5/13)), we can use the identities and properties of trigonometric functions. Let's break down the problem step by step:
1. Start with the expression cos(inverse tan(4/3) + inverse cos(5/13)).
2. Recall the identity for the sum of angles of cosine: cos(A + B) = cos(A)cos(B) - sin(A)sin(B).
3. Substitute A = inverse tan(4/3) and B = inverse cos(5/13) into the identity. We get:
cos(inverse tan(4/3) + inverse cos(5/13)) = cos(inverse tan(4/3))cos(inverse cos(5/13)) -
sin(inverse tan(4/3))sin(inverse cos(5/13)).
4. Use the definitions of inverse trigonometric functions to simplify further:
Let x = inverse tan(4/3). This implies tan(x) = 4/3, and we can solve for x using the inverse tan function on a calculator. The result is approximately 0.93 radians.
Let y = inverse cos(5/13). This implies cos(y) = 5/13, and we can solve for y using the inverse cos function on a calculator. The result is approximately 1.23 radians.
Therefore, the expression becomes:
cos(0.93 + 1.23) = cos(0.93)cos(1.23) - sin(0.93)sin(1.23).
5. Evaluate the trigonometric functions at the given angles:
cos(0.93) ≈ 0.599,
cos(1.23) ≈ 0.309,
sin(0.93) ≈ 0.804,
sin(1.23) ≈ 0.952.
6. Substitute these values back into the expression:
cos(0.93 + 1.23) ≈ 0.599 * 0.309 - 0.804 * 0.952.
7. Calculate the result:
cos(0.93 + 1.23) ≈ 0.184 - 0.768 ≈ -0.584.
Therefore, the exact value of cos(inverse tan(4/3) + inverse cos(5/13)) is approximately -0.584.
If the two angles are x and y,
cos(x+y) = cosx*cosy - sinx*siny
if tanx = 4/3,
cosx = 3/5
sinx = 4/5
if cosy = 5/13,
siny = 12/13
Now just plug and chug