cosx-sinx<1/2
square both sides
cos^2 -2sin*cos + sin^2<1/4
1-sin(2x)<1/4
sin(2x)<-3/4
2x>131deg
x>65.5 deg
2x<311
x<155 check that with a graph.
thnx
To solve the inequality cos(x) - sin(x) < 1/2, we can use algebraic methods along with trigonometric identities.
Step 1: Simplify the inequality if possible.
Since cos(x) - sin(x) cannot be simplified further, we can proceed to the next step.
Step 2: Rearrange the inequality to isolate the trigonometric terms.
cos(x) - sin(x) < 1/2
Step 3: Convert sin(x) and cos(x) to their equivalent expressions using the Pythagorean identity, sin^2(x) + cos^2(x) = 1.
Substituting cos^2(x) = 1 - sin^2(x) into the inequality, we get:
cos(x) - sin(x) < 1/2
cos(x) - √(1 - cos^2(x)) < 1/2
Step 4: Combine like terms.
Rearranging the inequality, we get:
cos(x) + cos(x)/2 < √(1 - cos^2(x))/2
(3/2)cos(x) < √(1 - cos^2(x))/2
Step 5: Square both sides of the inequality.
Square both sides of the inequality to eliminate the square root:
[(3/2)cos(x)]^2 < [(1 - cos^2(x))/2]^2
(9/4)cos^2(x) < (1 - cos^2(x))/4
Step 6: Simplify and rearrange the inequality.
Multiplying both sides of the inequality by 4 to eliminate the denominators, we get:
9cos^2(x) < 1 - cos^2(x)
10cos^2(x) < 1
Step 7: Solve for cos(x).
Subtracting cos^2(x) from both sides and rearranging, we have:
11cos^2(x) - 1 < 0
Step 8: Factor the equation.
Factoring the left side of the equation, we get:
(√11cos(x) - 1)(√11cos(x) + 1) < 0
Step 9: Solve the factorized equation.
Setting each factor less than zero, we get:
√11cos(x) - 1 < 0 and √11cos(x) + 1 > 0
Solving the first inequality:
√11cos(x) - 1 < 0
√11cos(x) < 1
cos(x) < 1/√11
Solving the second inequality:
√11cos(x) + 1 > 0
√11cos(x) > -1
cos(x) > -(1/√11)
Step 10: Write the final solution.
Combining the solutions from both inequalities, we have:
-(1/√11) < cos(x) < 1/√11
Therefore, the solution to the inequality cos(x) - sin(x) < 1/2 is -(1/√11) < cos(x) < 1/√11.