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Sagot :
Applying the trig formula: sin a = cos (90 - a)
cos (2x + 8) = sin (x + 37) = cos (90 - x - 37) = cos (53 - x)
Property of the cosine function -->
(
2
x
+
8
)
=
±
(
53
−
x
)
a. 2x + 8 = 53 - x
3x = 45
x
=
15
∘
b. 2x + 8 = - 53 + x
x
=
−
61
∘
For general answers, add
k
360
∘
Check by calculator.
x = 15 --> sin (x + 37) = sin 52 = 0.788
cos (2x + 8) = cos (38) = 0.788. Proved
x = - 61 --> sin (x + 37) = sin (- 24) = - sin 24 = - 0.407
cos (2x + 8) = cos (-122 + 8) = cos (- 114) = - 0.407. Proved
cos (2x + 8) = sin (x + 37) = cos (90 - x - 37) = cos (53 - x)
Property of the cosine function -->
(
2
x
+
8
)
=
±
(
53
−
x
)
a. 2x + 8 = 53 - x
3x = 45
x
=
15
∘
b. 2x + 8 = - 53 + x
x
=
−
61
∘
For general answers, add
k
360
∘
Check by calculator.
x = 15 --> sin (x + 37) = sin 52 = 0.788
cos (2x + 8) = cos (38) = 0.788. Proved
x = - 61 --> sin (x + 37) = sin (- 24) = - sin 24 = - 0.407
cos (2x + 8) = cos (-122 + 8) = cos (- 114) = - 0.407. Proved
The value of x which saisfies the equation sin(x+37)°=cos(2x+8)° is 135 or 15.
How to convert sine of an angle to some angle of cosine?
We can use the fact that:
[tex]\sin(\theta ^\circ) = \cos(90 - \theta^\circ)[/tex]
to convert the sine to cosine (but the angles won't stay same unless its 45 degrees).
For this case, we're specified the equation sin(x+37)°=cos(2x+8)°.
Converting sine to cosine, we get:
[tex]\cos(90 - x - 37)^\circ = \cos(2x + 8)^\circ\\[/tex]
Since cosine is a periodic function with period of [tex]360^\circ[/tex], thus, we get:
[tex]90 - x - 37= 2x + 8 +360 n[/tex]
where n = an integer (positive, negative, or zero).
or
[tex]90 - 37 - 8= 3x + 360 n\\\\x = \dfrac{45 - 360n}{3} = 15 - 120n[/tex]
This is the general solution of the considered equation.
Assuming that only principal values (from 0 to 360 degrees) angles are allowed, we need:
[tex]0 \leq x + 37 \leq 360\\\\and\\\\0 \leq 2x+8 \leq 360[/tex]
The first inequality gives:
[tex]-37 \leq x \leq 323[/tex]
The second inequality gives:
[tex]-4 \leq x \leq 176[/tex]
We need to satisfy both the inequalities, so the final boundaries on x are:
[tex]-4 \leq x \leq 176[/tex] (the minimum ones for which both inequalities stay true).
- n = -2 gives x = 255
- n = -1 gives x = 135
- n = 0 gives x = 15
- n = 1 gives x = -105
n < -2 gives x > 255, and n > 1 gives x < -105
So, values of n for which [tex]-4 \leq x \leq 176[/tex] is true are n = -1, or n = 0
Thus, x = either 135 or 15
sin
Thus, the value of x which saisfies the equation sin(x+37)°=cos(2x+8)° is 135 or 15
Learn more about sine to cosine conversion here:
https://brainly.com/question/1421592
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