evaluate the expressions below and leave them in radical form.
1. sin(pie symbol/6)csc(pie symbol/6)
2. sec(pie symbol/3)cos(pie symbol/3)+tan(pie symbol/3)cot(pie symbol/3)
1. sin (π/6) csc (π/6)
= sin (π/6) * 1/sin (π/6) = 1
2. sec(π/3)cos(π/3) + tan(π/3)cot(π/3)
I think for both questions, they want you to realize that if you multiply a trig function by its reciprocal function you get 1
= 1 + 1 = 2
To evaluate the expressions and leave them in radical form, we will need to use trigonometric identities and simplify the expressions.
1. We start with the expression sin(π/6) * csc(π/6).
First, we simplify the expression sin(π/6) using the trigonometric identity sin(π/6) = 1/2. So now we have (1/2) * csc(π/6).
Next, we simplify the expression csc(π/6) using the trigonometric identity csc(π/6) = 2/sqrt(3). We now have (1/2) * (2/sqrt(3)).
Multiplying these fractions, we get 2/(2 * sqrt(3)).
Simplifying further, we have 1/sqrt(3).
Therefore, sin(π/6) * csc(π/6) simplifies to 1/sqrt(3) or sqrt(3)/3 in radical form.
2. Now, we will evaluate the expression sec(π/3) * cos(π/3) + tan(π/3) * cot(π/3).
First, we simplify sec(π/3) using the trigonometric identity sec(π/3) = 2. So now we have 2 * cos(π/3) + tan(π/3) * cot(π/3).
Next, we simplify cos(π/3) using the trigonometric identity cos(π/3) = 1/2. So now we have 2 * (1/2) + tan(π/3) * cot(π/3).
Simplifying the expression further, we have 1 + tan(π/3) * cot(π/3).
Now, let's simplify tan(π/3) and cot(π/3). The trigonometric identities for these are tan(π/3) = sqrt(3) and cot(π/3) = 1/sqrt(3).
Substituting these values into our expression, we have 1 + sqrt(3) * (1/sqrt(3)).
This simplifies to 1 + 1 = 2.
Therefore, sec(π/3) * cos(π/3) + tan(π/3) * cot(π/3) evaluates to 2.