Evaluate each expression.csc225°\csc225\degreecsc225°

− 2 -\sqrt2 − 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

Evaluate each expression. csc 225 ° \csc225\degree csc 225°

In this problem we're asked to evaluate the expression and the expression that we're given here is the cosecant of 225 degrees. Now remember the cosecant is the reciprocal of sine so here I'm going to be taking one over that y value that represents sine on my unit circle. So coming down to my unit circle here at 225 degrees, I locate that y value, that sine value, of negative root two over two. Now one over this value is really just going to be flipping that fraction, so that gives me that this is a negative two over the square root of two having taken the reciprocal of y. Now of course we want to rationalize this denominator so in order to do that I can multiply this by the square root of two over the square root of two and I will end up getting a negative two times the square root of two on the top and then on the bottom I end up with two. Now I can further simplify this because those twos cancel out leaving me with the negative square root of two. So I'm done here and the cosecant of 225 is the negative square root of two. Thanks for watching and I'll see you in the next video.

Evaluate each expression. csc ⁡ 225 ° \csc225\degree csc 225°

− 2 -\sqrt2 − 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

− 2 2 -\frac{\sqrt2}{2} − 2 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

2 \sqrt2 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

12. Trigonometric Functions

Trigonometric Functions on the Unit Circle

Business Calculus

12. Trigonometric Functions

Trigonometric Functions on the Unit Circle

Struggling with Business Calculus?

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Multiple Choice

Evaluate each expression.
$\csc225\degree$

$1$

$-\frac{\sqrt2}{2}$

$\sqrt2$

$-\sqrt2$

Verified step by step guidance

Step 1: Recall the definition of the cosecant function. The cosecant (csc) of an angle is the reciprocal of the sine function. Mathematically, csc(θ) = 1/sin(θ).

Step 2: Determine the sine of 225°. The angle 225° is in the third quadrant, where sine is negative. The reference angle for 225° is 45°, so sin(225°) = -sin(45°).

Step 3: Use the known value of sin(45°). From trigonometric identities, sin(45°) = √2/2. Therefore, sin(225°) = -√2/2.

Step 4: Apply the reciprocal relationship to find csc(225°). Since csc(θ) = 1/sin(θ), substitute sin(225°) = -√2/2 into the formula. This gives csc(225°) = 1/(-√2/2).

Step 5: Simplify the expression for csc(225°). Dividing by a fraction is equivalent to multiplying by its reciprocal. Thus, csc(225°) = -2/√2. Simplify further to get the final result.

6:11m

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