Question

1. If sin y = sin 35, y≠35, what is the value of y? 2. If cos y = cos z = - cos 35, y≠z≠35, what are the values of y and z? 3. If sin y = sin x and x<90, y≠x, what is the value of y in terms of x? 4. If sin y = sin z = -sin x and x<90, y≠z≠x, what are the values of y and z in terms of x? 5. If cos y = cos x and x<90, y≠x, what are the value of y and z in terms of x? 6. If cos y = cos z = - cos x and x<90, y≠z≠x, what are the values of y and z in terms of x?

Answer 1

1. y = 145

2. y = 145 and z = 215

3. y = 180 - x

4. y = 180 + x and z = 360 - x

5. y = 360 - x

6. y = 180 - x and z = 180 + x

Explanation:

1. siny = sin35

Since sin35 is positive, siny is in the second quadrant.

So siny = sin(180 - 35) = sin35

siny = sin145 = sin35

Since siny = sin145, y = 145

2. cosy = cosz = -cos35

since -cos35 is negative in the second and third quadrant,

cosy = cos(180 - 35) = -cos35 and cosz = cos(180 + 35) = -cos35

cosy = cos145 = -cos35 and cosz = cos215 = -cos35

Since cosy = cos145 and cosz = cos215,

y = 145 and z = 215

3. If siny = sinx and x < 90,

since x < 90, sinx is in the first quadrant and siny is in the second quadrant since it is positive

siny = sin(180 - x) = sinx

Since siny = sin(180 - x)

y = 180 - x

4. siny = sinz = -sinx and x< 90

since -sinx is negative and sine is negative in the third and fourth quadrant respectively, siny and sinz are in the third and fourth quadrant,

siny = sin(180 + x) = -siny and sinz = sin(360 - x) = -sinx

Since siny = sin(180 + x) and sinz = sin(360 - x)

y = 180 + x and z = 360 - x

5. cosy = cosx and x < 90

Since cosx is positive in the first and fourth quadrant, and x is in the first quadrant,

cosy = cos(360 - x) = cosx

cosy = cos(360 - x)

y = 360 - x

6. cosy = cosz = -cosx and x< 90

since -cosx is negative in the second and third quadrant,

cosy = cos(180 - x) = -cosx and cosz = cos(180 + x) = -cosx

cosy = cos(180 - x) and cosz = cos(180 + x)

y = 180 - x and z = 180 + x