What is the general solution for (50.691)cos(50.691-c)+sin(50.691-c)=0 ?

The general solution for (50.691)cos(50.691-c)+sin(50.691-c)=0 is c=-0.70579…+180n

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(50.691)cos(50.691-c)+sin(50.691-c)=0

(50.691) cos (50.69 1^{\circ} - c) + sin (50.69 1^{\circ} - c) = 0

c = - 0.70579 \dots + 18 0^{\circ} n

Solution steps

(50.691) cos (50.69 1^{\circ} - c) + sin (50.69 1^{\circ} - c) = 0

Subtract

sin (50.69100 \dots^{\circ} - c)

from both sides

50.691 cos (0.88472 \dots - c) = - sin (- c + 50.69100 \dots^{\circ})

Square both sides

(50.691 cos (0.88472 \dots - c))^{2} = (- sin (- c + 50.69100 \dots^{\circ}))^{2}

Rewrite using trig identities

2569.577481 (0.63350 \dots cos (c) + 0.77374 \dots sin (c))^{2} = sin^{2} (- c + 50.69100 \dots^{\circ})

Subtract

sin^{2} (- c + 50.69100 \dots^{\circ})

from both sides

2569.577481 (0.63350 \dots cos (c) + 0.77374 \dots sin (c))^{2} - sin^{2} (- c + 0.88472 \dots) = 0

Factor

2569.577481 (0.63350 \dots cos (c) + 0.77374 \dots sin (c))^{2} - sin^{2} (- c + 0.88472 \dots) : (50.691 (0.63350 \dots cos (c) + 0.77374 \dots sin (c)) + sin (- c + 0.88472 \dots)) (50.691 (0.63350 \dots cos (c) + 0.77374 \dots sin (c)) - sin (- c + 0.88472 \dots))

(50.691 (0.63350 \dots cos (c) + 0.77374 \dots sin (c)) + sin (- c + 0.88472 \dots)) (50.691 (0.63350 \dots cos (c) + 0.77374 \dots sin (c)) - sin (- c + 0.88472 \dots)) = 0

Solving each part separately

50.691 (0.63350 \dots cos (c) + 0.77374 \dots sin (c)) + sin (- c + 0.88472 \dots) = 0 or 50.691 (0.63350 \dots cos (c) + 0.77374 \dots sin (c)) - sin (- c + 0.88472 \dots) = 0

50.691 (0.63350 \dots cos (c) + 0.77374 \dots sin (c)) + sin (- c + 0.88472 \dots) = 0 : c = arctan (- 0.85224 \dots) + 18 0^{\circ} n

50.691 (0.63350 \dots cos (c) + 0.77374 \dots sin (c)) - sin (- c + 0.88472 \dots) = 0 : c = arctan (- 0.78632 \dots) + 18 0^{\circ} n

Combine all the solutions

c = arctan (- 0.85224 \dots) + 18 0^{\circ} n, c = arctan (- 0.78632 \dots) + 18 0^{\circ} n

Verify solutions by plugging them into the original equation

c = arctan (- 0.85224 \dots) + 18 0^{\circ} n

Show solutions in decimal form

c = - 0.70579 \dots + 18 0^{\circ} n

sec (4 θ) = 2

tan^{2} (x) - 4 tan (x) + 1 = 0

sec (θ) + cos (θ) = sin (θ) tan (θ)

cos (x) = 0.98

tan (3 x + \frac{π}{2}) = - 1

What is the general solution for (50.691)cos(50.691-c)+sin(50.691-c)=0 ?
The general solution for (50.691)cos(50.691-c)+sin(50.691-c)=0 is c=-0.70579…+180n