Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number.
(tan(74°)-tan(14°))/(1+tan(74°)tan(14°))
=__________
Find it's exact value (not decimal form)
=__________
Note: A similar question(but with different degrees) had the answers for question part 1 and 2 as this
1). tan 30°
2). 1/square root{3}
1) Hmm, let's see if we can get creative with this one. How about we use the trigonometric identity tan(A - B) = (tan A - tan B) / (1 + tan A tan B)?
Using this identity, we can rewrite the expression as:
tan(74° - 14°) = tan(60°)
So the expression is equal to tan(60°).
2) The exact value of tan(60°) is √3. So the final answer is √3.
To write the expression as a trigonometric function of one number, we can use the subtraction formula for tangent:
tan(A - B) = (tan(A) - tan(B))/(1 + tan(A)tan(B))
So, we can rewrite the expression as:
(tan(74°) - tan(14°))/(1 + tan(74°)tan(14°)) = tan(74° - 14°)
Simplifying the angle in tangent, we have:
tan(74° - 14°) = tan(60°)
Now, let's find the exact value of tan(60°). We know that tan(60°) is equal to √3, so the exact value of the expression is:
= √3
To write the given expression as a trigonometric function of one number, we can use the tangent addition and subtraction formulas:
1. tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
2. tan(A - B) = (tan A - tan B) / (1 + tan A tan B)
First, let's plug in the values of A = 74° and B = 14° into the formula and simplify the expression:
(tan(74°) - tan(14°)) / (1 + tan(74°) * tan(14°))
Using the subtraction formula, we get:
(tan(74°) - tan(14°)) / (1 + tan(74°) * tan(14°)) =
= tan(74° - 14°)
= tan(60°)
So, the expression simplifies to tan(60°).
To find the exact value of tan(60°), we can use the properties of special triangles. In a 30-60-90 triangle, the tangent of the 60° angle is equal to the length of the side opposite the angle divided by the length of the side adjacent.
In a 30-60-90 triangle, the ratio of the side opposite the 60° angle to the side adjacent is √3:1. Therefore, the exact value of tan(60°) is √3.
Hence, the answers to the questions are:
1. The expression simplifies to tan(60°)
2. The exact value of the expression is √3.
recall that
tan(x-y) = (tanx-tany)/(1 + tanx tany)