Question

Find the exact values of the sine, cosine, and tangent of the angle.105° = 60° + 45°

Answer 1

since

`\begin{gathered} \sin(60)=(√(3))/(2) \\ \\ \cos(60)=(1)/(2) \\ \\ \tan(60)=√(3) \\ \\ \sin(45)=\cos(45)=(1)/(√(2)) \\ \\ \tan(45)=1 \end{gathered}`
`\begin{gathered} \Rightarrow\sin(105)=\sin(60+45)=\sin(60)\cos(45)+\sin(45)\cos(60)=(√(3))/(2)*(1)/(√(2))+(1)/(√(2))*(1)/(2) \\ \\ =(1)/(2)\sqrt{(3)/(2)}+(1)/(2)(1)/(√(2))=(1)/(2)((√(3)+1)/(√(2)))=(1)/(2)((√(6)+√(2))/(2))=(1)/(4)(√(6)+√(2)) \end{gathered}`

`\begin{gathered} \Rightarrow\cos(105)=\cos(60+45)=\cos(60)\cos(45)-\sin(60)\sin(45)=(1)/(2)*(1)/(√(2))-(√(3))/(2)*(1)/(√(2)) \\ \\ =(1)/(2√(2))-(1)/(2)(√(3))/(√(2))=(1)/(2)((1)/(√(2))-(√(3))/(√(2)))=(1)/(2)((1-√(3))/(√(2)))=(1)/(4)(√(2)-√(6)) \end{gathered}`
`\begin{gathered} \tan(105)=(\sin(105))/(\cos(105))=(√(2)+√(6))/(√(2)-√(6))=(√(2)+√(6))/(√(2)-√(6))*(√(2)+√(6))/(√(2)+√(6))=(2+2√(12)+6)/(2-6)=(8+4√(3))/(-4) \\ \\ =-2-√(3) \end{gathered}`