Answer:
We are given that s i n ( x ) = 1 2
. We can solve this equation for x by taking the sine inverse of both sides of the equation.
s i n − 1 ( s i n ( x ) ) = s i n − 1 ( 1 2 )
Simplifying the left-hand side gives the following:
x = s i n − 1 ( 1 2 )
Thus, x is the angle, such that
s i n ( x ) = 1 2 .
In trigonometry, there are special angles that have well known trigonometric values, and one of these angles is 30°. For an angle of measure 30°, we have the following:
s i n ( 30 ∘ ) = 1 2 c o s ( 30 ∘ ) = √ 3 2
Since
s i n ( 30 ∘ ) = 1 2 , we have that
s i n − 1 ( 1 2 ) = 30 ∘ , so x = 30°.
As we just saw, c o s ( 30 ∘ ) = √ 3 2
. To find tan(30°), we will use the trigonometric identity that
t a n θ =s i n θ c o s θ
. Thus, we have the following:
t a n( 30 ∘ ) = s i n ( 30 ∘ ) c o s ( 30 ∘ ) = 1 2 √ 3 2 = 1 2 ⋅ 2 √ 3 = 1 √ 3
We get that
t a n ( 30 ∘ ) = 1 √ 3
. Thus, all together, we have that if s i n ( x ) = 1 2 , then
c o s ( x ) = √ 3 2 and t a n ( x ) = 1 √ 3
.We are given that s i n ( x ) = 1 2
. We can solve this equation for x by taking the sine inverse of both sides of the equation.
s i n − 1 ( s i n ( x ) ) = s i n − 1 ( 1 2 )
Simplifying the left-hand side gives the following:
x = s i n − 1 ( 1 2 )
Thus, x is the angle, such that s i n ( x ) = 1 2 .
In trigonometry, there are special angles that have well known trigonometric values, and one of these angles is 30°. For an angle of measure 30°, we have the following:
s i n ( 30 ∘ ) =1 2 c o s ( 30 ∘ ) = √ 3 2 Since s i n ( 30 ∘ ) = 1 2 , we have that s i n − 1 ( 1 2 ) = 30 ∘ , so x = 30°.
As we just saw, c o s ( 30 ∘ ) = √ 3 2
. To find tan(30°), we will use the trigonometric identity that t a n θ = s i n θ c o s θ
. Thus, we have the following:
t a n ( 30 ∘ ) = s i n ( 30 ∘ ) c o s ( 30 ∘ ) = 1 2 √ 3 2 = 1 2 ⋅ 2 √ 3 = 1 √ 3
Explanation:
n trigonometry, the inverse sine function, denoted as
s i n − 1 x , is defined as the function that undoes the sine function. That is, s i n − 1 x is equal to the the angle, θ, such that s in θ = x , and s i n − 1 ( s i n ( θ )) = θ
. We can use this definition to determine the angle that corresponds to a specific sine value