Respuesta :
Solution:
As we know reference angle is smallest angle between terminal side and X axis.
As cosine 45 ° is always positive in first and fourth quadrant.
i.e CosФ, Cos (-Ф) or Cos(2π - Ф) have same value.
As, Cos 45°, Cos (-45°) or Cos ( 360° - 45°)= Cos 315°are same.
So, Angles that share the same Cosine value as Cos 45° have same terminal sides will be in Quadrant IV having value Either Cos (-45°) or Cos (315°).
Also, Cos 45° = Sin 45° or Sin 135° i.e terminal side in first Quadrant or second Quadrant.
You can use the definition of reference angles to find the reference angles of the given angle. Then use their values to evaluate the cosine of them and see if they equate to cos(45°)
The needed angles are:
[tex]360^\circ \times x + 315^\circ ; \: x \in \mathbb Z[/tex]
What is a reference angle of a given angle?
Think of reference angle as the minimum angle reaching from x axis to the terminal side of the given angle. Thus, if suppose the angle is 180 degrees, then it is overlapping on x axis, thus, the reference angle is 0.
If the angle is 135 degrees, we can reach it by shortcut from other side of x axis with only 45 degree walk. Thus, reference angle is 45 degrees.(see diagram attached below)
If its right angle, there is no choice, but only right angle to be as the reference angle.
Which trigonometric functions are positive in which quadrant?
- In first quadrant (0 < θ < π/2), all six trigonometric functions are positive.
- In second quadrant(π/2 < θ < π), only sin and cosec are positive.
- In the third quadrant (π < θ < 3π/2), only tangent and cotangent are positive.
- In fourth (3π/2 < θ < 2π = 0), only cos and sec are positive.
(this all positive negative refers to the fact that if you use given angle as input to these functions, then what sign will these functions will evaluate based on in which quadrant does the given angle lies.)
Using the above definitions, we get the reference angles of 45 degrees (from second third and fourth quadrant) within 360 degrees(as angles 360 degrees will make evaluate same cosines as of the angles having same terminal side of under 360 degrees) and their cosines as
[tex]180 - 45 = 135^\circ \text{\: (Second quadrant)}\\180 + 45 = 225^\circ \text{\: (Third quadrant)}\\360- 45 = 315^\circ \text{\: (Fourth quadrant)}[/tex]
Cosines of them are
[tex]cos(135^\circ) = cos(180 - 45) = -cos(45^\circ) = -\dfrac{1}{\sqrt{2}} \neq \dfrac{1}{\sqrt{2}} \\cos(225^\circ) = cos(180 + 45) = -cos(45^\circ) = -\dfrac{1}{\sqrt{2}} \neq \dfrac{1}{\sqrt{2}} \\\\cos(315^\circ) = cos(360 - 45) = cos(45^\circ) = \dfrac{1}{\sqrt{2}} = \dfrac{1}{\sqrt{2}} \\[/tex]
Thus, the reference angle(under 360 degrees) of 45 degrees which is not in first quadrant and have its cosine of same value as cos 45° is
315°
To get generalization, we can add multiples of full rotation which would give us family of such angles as [tex]360^\circ \times x + 315^\circ ; \: x \in \mathbb Z[/tex]
Thus,
The needed angles are:
[tex]360^\circ \times x + 315^\circ ; \: x \in \mathbb Z[/tex]
Learn more about reference angles here:
https://brainly.com/question/2697077
