The x-coordinate of the point at which the terminal side of the angle intersects the unit circle is defined as cos θ. Similarly, sin θ is the y-coordinate of that point.
The length of the leg on the y-axis is the y-coordinate of P, which is sin α. The length of the other leg is the x-coordinate of P, which is cos α. Keep in mind that P is in Quadrant II. Therefore, the x-coordinate is negative which means that cos α is negative.
P(x,y) = P(-cos α, sin α)We can find the value of α knowing that a half of a circle is 180^(∘) and the given angle has a measure of 165^(∘) in the counterclockwise direction.
α=180^(∘)-165^(∘) ⇔ α = 15^(∘)
We will now focus on the right triangle.
Now, we need to find the exact value for sin 15^(∘) which is equivalent to sin 165^(∘). To do this, we will rewrite 15^(∘) as 45^(∘)-30^(∘). Before we rewrite the given expression, let's recall the values of the three main trigonometric functions for the most important angles.
sin θ
cos θ
tan θ
θ =0^(∘)
0
1
0
θ =30^(∘)
1/2
sqrt(3)/2
sqrt(3)/3
θ =45^(∘)
sqrt(2)/2
sqrt(2)/2
1
θ =60^(∘)
sqrt(3)/2
1/2
sqrt(3)
θ =90^(∘)
1
0
-
θ =180^(∘)
0
- 1
0
θ =360^(∘)
0
1
0
Let's now recall the Angle Difference Identity for a sine function
sin(A-B)=sin A cos B-cos A sin B
We will use this identity to rewrite the given expression.
sin 45^(∘) cos 30^(∘) - cos 45^(∘) sin 30^(∘)
Next, we will use the table we constructed above to find the exact value of the expression.
sin 45^(∘) cos 30^(∘) - cos 45^(∘) sin 30^(∘)
=
sqrt(2)/2 (sqrt(3)/2)- sqrt(2)/2 (1/2)
Let's finally simplify the obtained expression!
