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The angle \theta_1θ 1 ​ theta, start subscript, 1, end subscript is located in Quadrant \text{II}IIstart text, I, I, end text, and \sin(\theta_1)=\dfrac{1}{4}sin(θ 1 ​ )= 4 1 ​ sine, (, theta, start subscript, 1, end subscript, ), equals, start fraction, 1, divided by, 4, end fraction . What is the value of \cos(\theta_1)cos(θ 1 ​ )cosine, (, theta, start subscript, 1, end subscript, )?

Respuesta :

Answer:

[tex]\cos(\theta_1)=-\dfrac{\sqrt{15}}{4}[/tex]

Step-by-step explanation:

The angle [tex]\theta_1[/tex] is located in Quadrant II.

[tex]\sin(\theta_1)=\dfrac{1}{4}[/tex]

From trigonometry, we know that:

[tex]\sin(\theta)=\dfrac{Opposite}{Hypotenuse}\\$Therefore:\\Opposite=1\\Hypotenuse=4\\Using Pythagorean theorem:\\Hypotenuse^2=Opposite^2+Adjacent^2\\4^2=1^2+Adjacent^2\\Adjacent^2=16-1\\Adjacent^2=15\\Adjacent=\sqrt{15}[/tex]

Now, in Quadrant II,

  • The x-axis is negative
  • The y-axis is positive

Therefore, the Adjacent angle to [tex]\theta_1 =-\sqrt{15}[/tex]

Therefore:

[tex]\cos(\theta_1)=\dfrac{Adjacent}{Hypotenuse}=\dfrac{-\sqrt{15}}{4}\\\\\cos(\theta_1)=-\dfrac{\sqrt{15}}{4}[/tex]