Respuesta :
From the fundamental law of trigonometric, we have
[tex] \sin^2(\theta) + \cos^2(\theta) = 1 \iff \cos(\theta) = \pm\sqrt{1-\sin^2(\theta)}[/tex]
Since [tex] 0 < \theta < \frac{\pi}{2} [/tex], the cosine must be positive, so we choose the positive solution:
[tex] \cos(\theta) = \sqrt{1-\dfrac{1}{9}} = \dfrac{\sqrt{8}}{3} [/tex]
Now, the rule for the double angle of the tangent states that
[tex]\tan(2\theta) = \dfrac{2\sin(\theta)\cos(\theta)}{\cos^2(\theta) - \sin^2(\theta)} = \dfrac{2\cdot\frac{1}{3}\cdot\frac{\sqrt{8}}{3}}{\frac{8}{9}-\frac{1}{9}} = \dfrac{\frac{2\sqrt{8}}{9}}{\frac{7}{9}} = \dfrac{2\sqrt{8}}{9}\cdot \dfrac{9}{7} = \dfrac{2\sqrt{8}}{7}[/tex]
:
c. 4√2 / 7 .
Step-by-step explanation:
Answer
sin Ф = 1/3.
cos Ф = √(1 - (1/3)^2) Note this will be the positive square root as
Ф is in the first quadrant.
sin 2Ф = 2 sin Ф cos Ф
= 2* 1/3 * √(1 - (1/3)^2)
= 2/3 * 2√2/3
= 4√2 / 9
cos 2Ф = 1 -2 sin^2 Ф = 1 - 2* (1/3)^2 = 7/9
tan 2 Ф = sin 2Ф / cos 2Ф = 4√2/ 9 / 7/9
= 4√2 / 9 * 9 / 7
= 4√2 / 7 (answer).
