Question

In a right triangle, 0 is an acute and sin 0=2/7. Evaluate the other five trigonometric functions of 0.

Answer 1

`\sin\theta=(2)/(7)\\\\\text{use}\ \sin^2\theta+\cos^2\theta=1\to\left((2)/(7)\right)^2+\cos^2\theta=1\\\\(4)/(49)+\cos^2\theta=1\ \ \ |-(4)/(49)\\\\\cos^2\theta=(45)/(49)\to\cos\theta=\sqrt{(45)/(49)}\\\\\cos\theta=(√(45))/(√(49))\\\\\cos\theta=(√(9\cdot5))/(7)\\\\\cos\theta=(3\sqrt5)/(7)`

`\text{use}\ \tan\theta=(\sin\theta)/(\cos\theta)\to\tan\theta=((2)/(7))/((3\sqrt5)/(7))=(2)/(7)\cdot(7)/(3\sqrt5)=(2)/(3\sqrt5)\cdot(\sqrt5)/(\sqrt5)\\\\=(2\sqrt5)/(3\cdot5)=(2\sqrt5)/(15)`

`\text{use}\ \cot\theta=(\cos\theta)/(\sin\theta)\to\cot\theta=((3\sqrt5)/(7))/((2)/(7))=(3\sqrt5)/(7)\cdot(7)/(2)=(3\sqrt5)/(2)`

`\text{use}\ \sec\theta=(1)/(\cos\theta)\to\sec\theta=(1)/((3\sqrt5)/(7))=(7)/(3\sqrt5)\cdot(\sqrt5)/(\sqrt5)=(7\sqrt5)/(3\cdot5)=(7\sqrt5)/(15)`

`\text{use}\ \csc\theta=(1)/(\sin\theta)\to\csc\theta=(1)/((2)/(7))=(7)/(2)`