1)tan Q = -3/4 Find cosQ
-3^2 + 4^2 = x^2
9+16 = sqrt 25 = 5
cos = ad/hy = -4/5
Am I correct?
2) Use the sum and difference identites
sin[x + pi/4] + sin[x-pi/4] = -1
sinx cospi/4 + cosxsin pi/4 + sinx cos pi/4 - cosx sin pi/4 = -1
2 sin x cos pi/4 =-1
cos pi/4 = sqr2/2
2sin^x(sqrt2/2) = -1
sin x = -sqrt2
x = 7pi/4 and 5pi/4
Am I correct?
-3^2 + 4^2 = x^2
9+16 = sqrt 25 = 5
cos = ad/hy = -4/5
Am I correct?
Yes, but that is only in quadrant 3.
Over in quadrant 4, say x = +4 and y = -3
then cos = 4/5
1) To find cosQ given that tanQ = -3/4, you can use the Pythagorean identity: tan^2Q + 1 = sec^2Q.
Since tanQ = -3/4, we can square both sides: (tanQ)^2 = (-3/4)^2. This gives us 9/16.
Now, we can substitute this value into the Pythagorean identity: (tanQ)^2 + 1 = sec^2Q. Plugging in the value we found, we get 9/16 + 1 = sec^2Q, which simplifies to 25/16 = sec^2Q.
Taking the square root of both sides, we get secQ = ±5/4.
Since secQ = 1/cosQ, we can find cosQ by taking the reciprocal of secQ: cosQ = ±4/5.
Therefore, the value of cosQ can be either -4/5 or 4/5, depending on the quadrant in which Q lies.
So, the answer should include both possibilities: cosQ = ±4/5.
2) To solve the equation sin[x + pi/4] + sin[x - pi/4] = -1 using the sum and difference identities, you can apply the formula for the sum of two sine functions: sin(A + B) = sinA cosB + cosA sinB and sin(A - B) = sinA cosB - cosA sinB.
Using these identities, we can rewrite the equation as:
sinx cos(pi/4) + cosx sin(pi/4) + sinx cos(pi/4) - cosx sin(pi/4) = -1.
Simplifying, we get:
2sinx cos(pi/4) - 2cosx sin(pi/4) = -1.
Recalling that cos(pi/4) = sqrt(2)/2 and sin(pi/4) = sqrt(2)/2, we substitute these values into the equation:
2sinx (sqrt(2)/2) - 2cosx (sqrt(2)/2) = -1.
simplifying further, we have:
sqrt(2) sinx - sqrt(2) cosx = -1.
Dividing the entire equation by sqrt(2), we get:
sinx - cosx = -1/sqrt(2).
Now, we can recall the equations for sin(pi/4) and cos(pi/4):
sin(pi/4) = 1/sqrt(2) and cos(pi/4) = 1/sqrt(2).
Therefore, we can rewrite the equation as:
sinx - cosx = -sin(pi/4).
Using the subtraction identity sinA - sinB = 2sin((A - B)/2) cos((A + B)/2), we can rewrite the equation as:
2sin((x - pi/4)/2)cos((x + pi/4)/2) = sin(pi/4).
Since sin(pi/4) = 1/sqrt(2), we substitute this value into the equation:
2sin((x - pi/4)/2)cos((x + pi/4)/2) = 1/sqrt(2).
Now, we can solve for x by isolating each sine and cosine term:
sin((x - pi/4)/2)cos((x + pi/4)/2) = 1/(2sqrt(2)).
Now, there are various possible values of x that satisfy this equation. To find them, you can set each sine and cosine term to be either 1 or -1, and solve for x. This will give you multiple solutions.
For example, if sin((x - pi/4)/2) = 1, then (x - pi/4)/2 = pi/2 or (x - pi/4)/2 = 3pi/2. Solving for x, we get x = 7pi/4 and x = 5pi/4.
Similarly, if sin((x - pi/4)/2) = -1, then (x - pi/4)/2 = -pi/2 or (x - pi/4)/2 = -3pi/2. Solving for x, we get x = -3pi/4 and x = -5pi/4.
Therefore, the values of x that satisfy the equation sin[x + pi/4] + sin[x - pi/4] = -1 are x = 7pi/4, x = 5pi/4, x = -3pi/4, and x = -5pi/4.
So, the answer should include all these solutions: x = 7pi/4, 5pi/4, -3pi/4, -5pi/4.