solve each equation for è, giving a general formula for all of the solutions:
6. tanè=-1
7. sin(è/2)=1/2
Please show step by step on how to solve these. Im not sure on how to do them. Thank you for your help!
tanx = -1
you need to know that:
tan pi/4 = 1
tan x is negative in QII and QIV
so, tanx = -1 if x = pi-pi/4 = 3pi/4 or 2pi-pi/4 = 7pi/4
sin(x/2) = 1/2
sin pi/6 = 1/2
sin x is positive in QI and QII
so, sinx = 1/2 if x = pi/6 or pi-pi/6 = 5pi/6
Oops, that would be x/2 = pi/6 or 5pi/6
so, x = pi/3 or 5pi/3
To solve equation 6, tanè = -1, for è, we can use the properties of the tangent function and the unit circle. Here are the steps:
Step 1: Recall that the tangent function is defined as the ratio of the sine and cosine functions: tan(è) = sin(è) / cos(è).
Step 2: Rewrite the given equation in terms of sine and cosine: sin(è) / cos(è) = -1.
Step 3: Multiply both sides of the equation by cos(è) to eliminate the denominator: sin(è) = -cos(è).
Step 4: Rewrite sin(è) in terms of the sine function's Pythagorean identity: sin(è) = √(1 - cos²(è)).
Step 5: Substitute √(1 - cos²(è)) for sin(è) in the equation: √(1 - cos²(è)) = -cos(è).
Step 6: Square both sides of the equation to eliminate the square root: 1 - cos²(è) = cos²(è).
Step 7: Expand and simplify the equation: 1 = cos²(è) + cos²(è) = 2cos²(è).
Step 8: Divide both sides of the equation by 2: cos²(è) = 1/2.
Step 9: Take the square root of both sides of the equation: cos(è) = ±√(1/2) = ±(1/√2).
Step 10: Simplify the square root: cos(è) = ±(√2/2).
Step 11: Determine the angles that have a cosine value of √2/2. From the unit circle, we find that cos(45°) = √2/2 and cos(225°) = -√2/2.
Therefore, the solutions for è in the equation tanè = -1 are:
è = 45° + nπ and è = 225° + nπ, where n is an integer.
To solve equation 7, sin(è/2) = 1/2, for è, we can use the properties of the sine function and the unit circle. Here are the steps:
Step 1: Recall that the half-angle identity for sine is given by: sin(è/2) = ±√((1 - cos(è))/2).
Step 2: Rewrite the given equation as: ±√((1 - cos(è))/2) = 1/2.
Step 3: Square both sides of the equation to eliminate the square root: (1 - cos(è))/2 = (1/2)² = 1/4.
Step 4: Multiply both sides of the equation by 2 to isolate the term with cos(è): 1 - cos(è) = 1/4 * 2 = 1/2.
Step 5: Subtract 1/2 from both sides of the equation: -cos(è) = 1/2 - 1 = -1/2.
Step 6: Multiply both sides of the equation by -1 to isolate the term with cos(è): cos(è) = 1/2.
Step 7: Determine the angles that have a cosine value of 1/2. From the unit circle, we find that cos(60°) = 1/2 and cos(300°) = 1/2.
Therefore, the solutions for è in the equation sin(è/2) = 1/2 are:
è/2 = 60° + 360°n and è/2 = 300° + 360°n, where n is an integer.
To obtain the general formula for all solutions, we can multiply both sides of each equation by 2 and solve for è:
For the first equation (tanè = -1), the general formula for all solutions is:
è = 2(45° + nπ) and è = 2(225° + nπ), where n is an integer.
For the second equation (sin(è/2) = 1/2), the general formula for all solutions is:
è = 2(60° + 360°n) and è = 2(300° + 360°n), where n is an integer.