Question

Find the value of cos 60°+ √3 cos 30°+ sin 30°.
Find the value of tan 30°+ cot 30°+ sin 30°. ​

Answer 1

`\cos 60^(\circ)+ √(3) \cos 30^(\circ)+ \sin 30^(\circ)=(5)/(2)`

`\tan 30^(\circ)+\cot30^(\circ)+\sin30^(\circ)=(3+8√(3))/(6)`

Explanation:

Question 1

To find the value of the given trigonometric expression, we can use the unit circle to first find the values of cos 60°, cos 30° and sin 30°.

In the unit circle, the cosine of the angle is the x-coordinate of a point on the circle, and the sine of the angle is the y-coordinate of that point.

Therefore, reading from the unit circle (attached):

`\boxed{\begin{minipage}{2.3 cm}$\cos 60^(\circ)=(1)/(2)$\\\\\\$\cos 30^(\circ)=(√(3))/(2)$\\\\\\$\sin 30^(\circ)=(1)/(2)$\\\end{minipage}}`

Substitute these values into the expression and solve:

`\begin{aligned}\cos 60^(\circ)+ √(3) \cos 30^(\circ)+ \sin 30^(\circ)&=(1)/(2)+√(3) \cdot (√(3))/(2)+(1)/(2)\\\\&=(1)/(2)+(3)/(2)+(1)/(2)\\\\&=(1+3+1)/(2)\\\\&=(5)/(2)\end{aligned}`

`\hrulefill`

Question 2

Use the following trigonometric identities to rewrite tan 30° and cot 30° in terms of sine and cosine.

`\boxed{\begin{minipage}{4 cm}\underline{Trigonometric identities}\\\\$\tan x=(\sin x)/(\cos x)$\\\\\\$\cot x=(\cos x)/(\sin x)$\\\\\end{minipage}}`

`\tan 30^(\circ)+\cot30^(\circ)+\sin30^(\circ)=(\sin30^(\circ))/(\cos30^(\circ))+(\cos30^(\circ))/(\sin30^(\circ))+\sin30^(\circ)`

Now use the unit circle to find the values of sin 30° and cos 30°:

`\boxed{\begin{minipage}{2.3 cm}$\sin 30^(\circ)=(1)/(2)$\\\\\\$\cos 30^(\circ)=(√(3))/(2)$\\\end{minipage}}`

Substitute these values into the expression and solve:

`\begin{aligned}\tan 30^(\circ)+\cot30^(\circ)+\sin30^(\circ)&=(\sin30^(\circ))/(\cos30^(\circ))+(\cos30^(\circ))/(\sin30^(\circ))+\sin30^(\circ)\\\\&=((1)/(2))/((√(3))/(2))+((√(3))/(2))/((1)/(2))+(1)/(2)\\\\&=(1)/(2)\cdot(2)/(√(3))+(√(3))/(2)\cdot(2)/(1)+(1)/(2)\\\\&=(1)/(√(3))+(√(3))/(1)+(1)/(2)\\\\&=(1 \cdot √(3))/(√(3)\cdot √(3))+(√(3))/(1)+(1)/(2)\\\\\end{aligned}`

`\begin{aligned}&=(√(3))/(3)+(√(3))/(1)+(1)/(2)\\\\&=(√(3)\cdot 2)/(3\cdot 2)+(√(3)\cdot 6)/(1\cdot 6)+(1\cdot 3)/(2\cdot 3)\\\\&=(2√(3))/(6)+(6√(3))/(6)+(3)/(6)\\\\&=(2√(3)+6√(3)+3)/(6)\\\\&=(8√(3)+3)/(6)\\\\&=(3+8√(3))/(6)\end{aligned}`