Use a Linear Approximation to estimate sin(64∘)−sin(60∘)≈

To estimate sin(64∘)−sin(60∘) using linear approximation, follow these steps:

Step 1: Convert the angles from degrees to radians.
The conversion formula is:
radians = degrees * (π/180)

For 64∘:
64 * (π/180) = 1.11701072 radians

For 60∘:
60 * (π/180) = 1.04719755 radians

Step 2: Use the linear approximation formula:
f(x + Δx) ≈ f(x) + f'(x) * Δx

In this case, the function is sin(x), and Δx = 1.11701072 - 1.04719755 = 0.06981317 radians.

Step 3: Calculate sin(x) and its derivative at x = 1.04719755 radians.

Using a calculator:
sin(1.04719755) ≈ 0.86602540378
cos(1.04719755) ≈ 0.5

Step 4: Substitute the values into the linear approximation formula:
sin(1.11701072) ≈ sin(1.04719755) + cos(1.04719755) * 0.06981317

Step 5: Calculate the result.
sin(1.11701072) ≈ 0.86602540378 + 0.5 * 0.06981317

Simplifying:
sin(1.11701072) ≈ 0.86602540378 + 0.034906585

The estimated value of sin(64∘)−sin(60∘) is approximately 0.900931989.

To estimate sin(64∘) - sin(60∘) using a linear approximation, we can use the fact that for small angles, sinθ is approximately equal to θ.

Step 1: Convert the angles from degrees to radians.
To work with trigonometric functions, we need to convert the angles from degrees to radians. We know that π radians is equivalent to 180 degrees, so we can use the following formula to convert degrees to radians:
radians = degrees × (π/180)

For 64 degrees:
radians = 64 × (π/180)

For 60 degrees:
radians = 60 × (π/180)

Step 2: Apply the linear approximation formula.
The linear approximation formula states that for small values of θ, sinθ ≈ θ. Therefore, we can approximate sin(64∘) as 64∘ radians and sin(60∘) as 60∘ radians.

Step 3: Calculate the approximation.
sin(64∘) - sin(60∘) ≈ (64∘ - 60∘) radians

Now we can simplify the expression:
sin(64∘) - sin(60∘) ≈ 4∘ radians

Finally, convert the radians back to degrees:
4∘ radians ≈ 4 radians x (180/π) ≈ 229.18 degrees (rounded to two decimal places)

Therefore, sin(64∘) - sin(60∘) is approximately equal to 229.18 degrees.

if y=sin x

y' = cos x

To approximate sin(x) at x=60°, use the tangent line at (π/3,√3/2) where the slope is 1/2

∆y = 1/2 ∆x = (1/2)(1.117-1.047) = 0.035