Trigonometry question

Question

Find the indicated trigonometric value in the specific quadrant:

function: csc theta = -11/4

quadrant: III

trigonometric value: tan theta

Answer 1

cscθ = -11/4, so sinθ = -4/11

tanθ = sinθ / cosθ

In quadrant 3, cosθ < 0. So, cosθ = -√(1 - sin²θ) = -√105 / 11

Therefore, tanθ = sinθ / cosθ = 4 / √105

Answer 2

csc=1/sin; =-11/4 or -2.75

1/sinx=-2.75

sinx*-2.75=1

sinx=-.3636

x=21.32°

Answer 3

For this example, let's say theta is "©".

SOH CAH TOA... Sine = Opposite over Hypothenuse, Cosine = Adjacent over Hypothenuse, Tangent Opposite over Adjacent.

csc© = 1/sin©

tan© = sin©/cos©

In Quadrant lll, sin and cos have (-) values.

Refer to a Unit Circle. I would draw it out but I'm still learning the video answer.

csc© = -(11/4)... sin© = -(4/11)... cos© = unknown... tan© = sin©/cos©

*Refer to the top for how I got the sin value.

Draw a triangle in Quadrant lll, so you have a line going left, down, then connect those to lines. The two sides (not the hypothenuse) will be negative values. The line going down = -4... The hypothenuse is 11. Solve for the missing side.

Hypothenuse can never be negative #. Hypothenuse will always be the largest number.

*SOH = Sine = Opposite side over Hypothenuse... Therefore the line going down = -4.

To solve for the missing side, use Pythagorean Theroem.

a^2 + b^2 = c^2

In our case, a = side for sin... b = side for cos, c = hypothenuse.

Looks like this:

(-4)^2 + (b)^2 = (11)^2...

b = √105

tan© = sin©/cos©...

a/b...

(-4)/(-√105)...

(-) # divided by (-) # = (+) number. Same thing for when you are multiplying #s.

*NEVER have square root (√) as the denominator (bottom number).

√105*(-4)/√105*(√105)...

ANSWER: tan© = (4*√105)/105

Questions? my e-mail is [email protected]