a sin75^(∘)=sqrt(6)+sqrt(2)/4
cos75^(∘)= sqrt(6)-sqrt(2)/4
B
b sin75^(∘)=sqrt(6)+sqrt(2)/4
cos75^(∘)= sqrt(6)-sqrt(2)/4
Practice makes perfect
a We are asked to use Sum Formulas to find the exact values of sin75^(∘) and cos75^(∘). Let's begin by writing 75^(∘) as the sum of two notable angles.
75^(∘)=45^(∘)+30^(∘)
Knowing this, we can use the corresponding sum formula. Let's do each one at a time!
Finding sin75^(∘)
Let's take a look at the Sum Formula for sine.
sin(a+b)=sinacosb+cosasinb
Let's use this formula to find sin75^(∘).
b We are now asked to use difference formulas to find the exact values of sin75^(∘) and cos75^(∘). Notice that we have to find two differences!
75^(∘)= 90^(∘) - 15^(∘) ⇓
75^(∘) = 90^(∘) - (45^(∘)-30^(∘))
We will need to find the values of sin15^(∘) and cos15^(∘) in order to use difference formulas. Let's find each one at a time.
Finding sin15^(∘)
We will use the Difference Formula for sine.
sin(a-b)=sinacosb-cosasinbWe found that 15^(∘)=45^(∘)-30^(∘). Let's put this into the Difference Formula!
