a) Solve for angle theta given that 0 < (theta) < 2 pie       cos2(theta) = 0  b) state the general solution (for all theta) for the equation tan(theta) = -1  c) Simlify: cos(pie/2 -...

a) Solve for angle theta given that 0 < (theta) < 2 pie      

cos2(theta) = 0

 

b) state the general solution (for all theta) for the equation tan(theta) = -1 

c) Simlify: cos(pie/2 - (theta)) + cos(pie-(theta)) + cos(3pie/2 - (theta)) + cos(2pie - (theta))

 

 

Asked on by maheen100

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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a) `cos 2 theta = 0 =gt 2 theta = cos^(-1) 0 `

2 theta = pi/2 or 2 theta = 3pi/2

`theta = pi/4or theta = 3pi/4`

`b) tan theta = -1`

`theta = tan^(-1)(-1) + n*pi`

`theta = -pi/4 + n*pi`

Hence, the general solution to the given equation is `theta = -pi/4 + n*pi.`

c) You should use the identity:

`cos(a - b) = cos a*cos b + sin a*sin b`

`cos(pi/2 - theta) = cos (pi/2)*cos theta + sin (pi/2)*sin theta`

`cos(pi/2 - theta) = 0*cos theta + 1*sin theta`

`cos(pi/2 - theta) = sin theta`

`cos(pi - theta) = cos (pi)*cos theta + sin (pi)*sin theta ` `cos(pi - theta) = -cos theta`

`cos(3pi/2 - theta) = cos (3pi/2)*cos theta + sin (3pi/2)*sin theta`

`cos(3pi/2 - theta) = -sin theta`

`cos(2pi - theta) = cos (2pi)*cos theta + sin (2pi)*sin theta`

`cos(2pi - theta) = cos theta `

You need to substitute the results in expression such that:

`sin theta - cos theta - sin theta + cos theta = 0`

Hence, evaluating the expression  `cos(pi/2 - theta) + cos(pi - theta)+ cos(3pi/2 - theta) + cos(2pi - theta) = 0` .

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