The value of the expression [tex]2\cos(t) + \tan^2(t)[/tex] is 4 and the exact value of [tex]\sin^{-1}(\sin(\frac{5\pi}{3}))[/tex] is [tex]\frac{5\pi}{3}[/tex]
How to determine the trigonometry expression?
The point on the unit circle is given as:
[tex]P = (\frac 12, \frac{\sqrt{3}}2)[/tex]
A point on a unit circle is represented as: (x,y), such that:
cos(t) = x and sin(t) = y.
This means that:
[tex]\cos(t) = \frac 12[/tex]
[tex]\sin(t) = \frac{\sqrt 3}{2}[/tex]
Calculate tan(t) using:
[tex]\tan(t) = \frac{\sin(t)}{\cos(t)}[/tex]
So, we have:
[tex]\tan(t) = \frac{\frac{\sqrt 3}{2}}{1/2}[/tex]
Evaluate
[tex]\tan(t) = \sqrt 3[/tex]
The expression is then calculated as:
[tex]2\cos(t) + \tan^2(t) = 2 * \frac12 + (\sqrt 3)^2[/tex]
Evaluate each term
[tex]2\cos(t) + \tan^2(t) = 1 + 3[/tex]
Evaluate the sum
[tex]2\cos(t) + \tan^2(t) = 4[/tex]
Hence, the value of the expression [tex]2\cos(t) + \tan^2(t)[/tex] is 4
How to solve the arcsin expression?
The expression is given as:
[tex]\sin^{-1}(\sin(\frac{5\pi}{3}))[/tex]
As a general rule, the arc sine of sine x is x.
This means that:
[tex]\sin^{-1}(\sin(\frac{5\pi}{3})) = \frac{5\pi}{3}[/tex]
Hence, the exact value of [tex]\sin^{-1}(\sin(\frac{5\pi}{3}))[/tex] is [tex]\frac{5\pi}{3}[/tex]
Read more about trigonometry expressions at:
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