solve the given equation
cos(θ) − sin(θ) = 1
If theta is zero then cos(θ) is one and sin(θ)is zero
what about when sin=-1 and cos=0?
to make sure, let's work it out.
Note that cos π/4 = sin π/4 = 1/√2
cosθ - sinθ
1/√2 cosθ - 1/√2 sinθ = 1/√2
cos(θ+π/4) = 1/√2
θ+π/4 = π/4 or 7π/4
θ = 0 or 3π/2
To solve the equation cos(θ) - sin(θ) = 1, we need to use some trigonometric identities and algebraic techniques. Here's how you can approach it:
Step 1: Rearrange the equation to get all the trigonometric terms on one side and the constant term on the other side:
cos(θ) - sin(θ) - 1 = 0
Step 2: Use the trigonometric identity cos(θ) = sin(π/2 - θ) to replace cos(θ) in the equation:
sin(π/2 - θ) - sin(θ) - 1 = 0
Step 3: Apply the trigonometric identity sin(A) - sin(B) = 2cos((A + B)/2)sin((A - B)/2) to combine the terms:
2cos((π/2 - θ + θ)/2)sin((π/2 - θ - θ)/2) - 1 = 0
Simplifying further:
2cos(π/4)sin(-θ/2) - 1 = 0
Step 4: Use the trigonometric identity cos(π/4) = sin(π/4) = 1/√2:
2(1/√2)sin(-θ/2) - 1 = 0
√2sin(-θ/2) - 1 = 0
Step 5: Multiply both sides of the equation by √2 to eliminate the square root:
sin(-θ/2) = √2/2
Step 6: Take the inverse sine of both sides to find the value of -θ/2:
-θ/2 = arcsin(√2/2)
Step 7: Solve for θ by multiplying both sides by -2 and simplifying:
θ = -2 * arcsin(√2/2)
Finally, use a calculator to find the numerical value for θ.