Question

Using the attached link below.

A. Find sin x and csc y

B. Find tan x and cot y

C. Find cos x and sec y

D. And if sin theta = 2/3, find the values of cos theta and tan theta

Answer 1

`\text{Use the Pythagorean theorem:}\\\\r-hypotenuse\\\\r^2=7^2+5^2\\\\r^2=49+25\\\\r^2=74\to r=√(74)`

`\sin=(opposite)/(hypotenuse)\\\\\cos=(adjacent)/(hypotenuse)\\\\\tan=(opposite)/(adjacent)\\\\\cot=(adjacent)/(opposite)\\\\\text{We have}\\\\for\ the\ angle\ y:\\\text{opposite = 7}\\\text{adjacent = 5}\\\text{hypotenuse = }\ √(74)\\\\for\ the\ angle\ x:\\\text{opposite = 5}\\\text{adjacent = 7}\\\text{hypotenuse = }\ √(74)`

`\csc x=(1)/(\sin x)\\\\\sec x=(1)/(\cos x)`

`A.\\\\\sin x=(5)/(√(74))=(5√(74))/(74)\\\\\csc y=(1)/((7)/(√(74)))=(√(74))/(7)\\\\B.\\\\\tan x=(5)/(7)\\\\\cot y=\dfrc{5}{7}\\\\C.\\\\\cos x=(7)/(√(74))=(7√(74))/(7)\\\\\sec y=(1)/((5)/(√(74)))=(√(74))/(5)`

`D.\\\sin\theta=(2)/(3)\\\\\sin^2\theta+\cos^2\theta=1\to\left((2)/(3)\right)^2+\cos^2\theta=1\\\\(4)/(9)+\cos^2\theta=1\qquad\text{subtract}\ (4)/(9)\ \text{from both sides}\\\\\cos^2\theta=(5)/(9)\to\cos\theta=\sqrt{(5)/(9)}\to\cos\theta=(\sqrt5)/(3)\\\\\tan\theta=(\sin\theta)/(\cos\theta)\to\tan\theta=((2)/(3))/((\sqrt5)/(3))=(2)/(3)\cdot(3)/(\sqrt5)=(2)/(\sqrt5)=(3\sqrt5)/(5)`