Construct an appropriate triangle to complete the table. (0 ≤ 𝜃 ≤ 90°, 0 ≤ 𝜃 ≤ 𝜋/2)

Question

Function: csc

 𝜃 (deg): ______

 𝜃 (rad): ______



Function value: √2

Answer 1

csc(theta) = sqr2 = 1/sin(theta)

sin(theta) = 1/sqr2 = sqr2/2 = about 1.414/2 = 0.707

the angle whose sine is sqr2/2 is 45 degrees or pi/4 radians

a right triangle is formed with sides 1, 1 and sqr2. It's a common angle in trigonometry that people generally recognize. Or use a calculator with an inverse trig function. theta = sin^-1(.707) = 45 degrees. If you used 0.707, you'd get 44.99 degrees. You'd need to put in sqr2/2 to get the exact solution of 45 degrees.

the limits on the angle place it in quadrant I, otherwise, theta could have been 135 degrees & 3pi/4

as well as 45 and pi/4. With each angle also plus n360 or 2npi where n=any integer.

Answer 2

if csc=square root of 2 then sin=1/csc, is square root of 2/2=.707

a valid angle is 45 degrees, pi/4 radians.

as for triangle, draw a line from origin to x=square root of 2/2, y=square root of 2/2 (the point x,y)

and from that point a vertical line, the x-axis is third side. you now have an isosceles 90, 45, 45 degree triangle

with a hypotenuse length of one unit.