What is the general solution for cos(w)=0.75sin(90-w) ?

The general solution for cos(w)=0.75sin(90-w) is w=90+360n,w=270+360n

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cos(w)=0.75sin(90-w)

Solution

cos (w) = 0.75 sin (9 0^{\circ} - w)

Solution

w = 9 0^{\circ} + 36 0^{\circ} n, w = 27 0^{\circ} + 36 0^{\circ} n

Radians

w = \frac{π}{2} + 2 πn, w = \frac{3 π}{2} + 2 πn

Solution steps

cos (w) = 0.75 sin (9 0^{\circ} - w)

Rewrite using trig identities

cos (w) = 0.75 sin (9 0^{\circ} - w)

Rewrite using trig identities

sin (9 0^{\circ} - w)

Use the Angle Difference identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= sin (9 0^{\circ}) cos (w) - cos (9 0^{\circ}) sin (w)

Simplify

sin (9 0^{\circ}) cos (w) - cos (9 0^{\circ}) sin (w) : cos (w)

sin (9 0^{\circ}) cos (w) - cos (9 0^{\circ}) sin (w)

sin (9 0^{\circ}) cos (w) = cos (w)

sin (9 0^{\circ}) cos (w)

Simplify

sin (9 0^{\circ}) : 1

sin (9 0^{\circ})

Use the following trivial identity:

sin (9 0^{\circ}) = 1

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

= 1 \cdot cos (w)

Multiply:

1 \cdot cos (w) = cos (w)

= cos (w)

cos (9 0^{\circ}) sin (w) = 0

cos (9 0^{\circ}) sin (w)

Simplify

cos (9 0^{\circ}) : 0

cos (9 0^{\circ})

Use the following trivial identity:

cos (9 0^{\circ}) = 0

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= 0

= 0 \cdot sin (w)

Apply rule

0 \cdot a = 0

= 0

= cos (w) - 0

cos (w) - 0 = cos (w)

= cos (w)

= cos (w)

cos (w) = 0.75 cos (w)

cos (w) = 0.75 cos (w)

Subtract

0.75 cos (w)

from both sides

0.25 cos (w) = 0

Multiply both sides by

100

0.25 cos (w) = 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

0.25 cos (w) \cdot 100 = 0 \cdot 100

Refine

25 cos (w) = 0

25 cos (w) = 0

Divide both sides by

25

25 cos (w) = 0

Divide both sides by

25

\frac{25 cos ( w )}{25} = \frac{0}{25}

Simplify

cos (w) = 0

cos (w) = 0

General solutions for

cos (w) = 0

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

w = 9 0^{\circ} + 36 0^{\circ} n, w = 27 0^{\circ} + 36 0^{\circ} n

w = 9 0^{\circ} + 36 0^{\circ} n, w = 27 0^{\circ} + 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for cos(w)=0.75sin(90-w) ?
The general solution for cos(w)=0.75sin(90-w) is w=90+360n,w=270+360n