Answer:
sin Ф = 3/√13; cos Ф = 2/√13; and tan Ф = 3/2
Step-by-step explanation:
Let's assume we're limiting ourselves to Quadrant I.
Start with the tangent function. tan Ф = opp / adj.
In this case opp = 3 and adj = 2.
The length of the hypotenuse is found using the Pythagorean Theorem and is √(3² + 2²) = √13.
Then sin Ф = opp / hyp = 3/√13 or 3√13/13
and
cos Ф = adj / hyp = 2/√13 or 2√13/13
and (as before)
tan Ф = opp / adj = 3/2
sin(θ) is approximately 0.832
cos(θ) is approximately 0.555
tan(θ) is 1.5
The given parameters are;
The location of point (2, 3) = The terminal side of angle θ in standard position
The required parameters;
sin of θ, cosine of θ, and tangent of θ
Strategy;
Draw angle θ on the coordinate plane based on definition showing point (2, 3) on the terminal side and find the required trigonometric ratio
Standard position is the position of an angle that has the vertex at the
origin, the fixed side of the angle is on the x-axis and the terminal side
which defines the angle is drawn relative to the initial fixed side to form
the given angle either clockwise or anticlockwise
We have the following trigonometric ratios with regards to the reference angle;
[tex]sin\angle X = \dfrac{Opposite \ leg \ length}{Hypotenuse \ length}[/tex]
The hypotenuse length = √(2² + 3²) = √13
Therefore;
[tex]\mathbf{sin( \theta)} = \dfrac{3 - 0}{\sqrt{13} }= \dfrac{3}{\sqrt{13} } = \mathbf{\dfrac{3 \cdot \sqrt{13} }{13}}[/tex]
[tex]\mathbf{sin( \theta)} \ is \ \mathbf{\dfrac{3 \cdot \sqrt{13} }{13}} \approx 0.832[/tex]
[tex]cos\angle X = \dfrac{Adjacent\ leg \ length}{Hypotenuse \ length}[/tex]
Therefore
[tex]\mathbf{cos( \theta)} = \dfrac{2 - 0}{\sqrt{13} }= \dfrac{2}{\sqrt{13} } =\mathbf{ \dfrac{2 \cdot \sqrt{13} }{13 }}[/tex]
[tex]\mathbf{cos( \theta)} \ is \ \mathbf{ \dfrac{2 \cdot \sqrt{13} }{13 }} \approx 0.555[/tex]
[tex]tan\angle X = \dfrac{Opposite \ leg \ length}{Adjacent\ leg \ length}[/tex]
The hypotenuse length = √(2² + 3²) = √13
Therefore;
[tex]\mathbf{tan( \theta)} = \dfrac{3 - 0}{2 - 0 } \mathbf{=\dfrac{3}{2 }}[/tex]
tan(θ) = 1.5
Learn more bout trigonometric ratios here;
https://brainly.com/question/17072886
