Find the exact value of the given expression.
(tan17pi/12 - tan pi/4) / (1+tan 17pi/12 tan pi/4)
can you help with this? i got the answer 1/ squareroot 3
I assume you used:
tan(A - B) = (tanA - tanB)/( 1 + tanAtanB)
then (tan17pi/12 - tan pi/4) / (1+tan 17pi/12 tan pi/4)
= tan(17π/12 - π/4)
= tan( 7π/6) or tan 210° ---> quad III
= 1/√3
You are correct!
To find the exact value of the given expression:
1. Start by simplifying the expression step by step.
2. Recall the trigonometric identity: tan(A - B) = (tan A - tan B) / (1 + tan A tan B).
3. Apply the identity to the first term of the expression:
tan(17π/12 - π/4) = (tan(17π/12) - tan(π/4)) / (1 + tan(17π/12) tan(π/4))
4. Use the values of the special angles:
tan(17π/12) = tan(π + 5π/12) = -tan(5π/12)
tan(5π/12) = tan(π/3 + 2π/12) = tan(π/3 + π/6) = tan(2π/3) = -√3
5. Similarly, tan(π/4) = 1.
6. Substitute these values into the expression:
= (-(-√3) - 1) / (1 + (-√3)(1))
= (√3 - 1) / (1 - √3)
= (√3 - 1)/(1 - √3) * (1 + √3)/(1 + √3)
= (√3 + √3 - 1 - √3) / (1 - 3)
= (√3 - 1) / -2
= (1 - √3) / 2
Therefore, the exact value of the given expression is (1 - √3) / 2.
Sure, I can help you with that.
To find the exact value of the expression, we'll use trigonometric identities to simplify it. Let's break it down step by step:
1. Start by simplifying the numerator, (tan(17π/12) - tan(π/4)):
We know that tan(π/4) equals 1. Therefore, our expression becomes:
(tan(17π/12) - 1).
2. Next, let's use the identity tan(x + π) = tan(x):
The angle 17π/12 is equivalent to (17π/12 + π). So we can rewrite the expression as:
(tan((17π/12) + π) - 1).
3. By applying the identity tan(x + π) = tan(x), the expression simplifies to:
(tan(5π/12) - 1).
4. Now, let's focus on the denominator, (1 + tan(17π/12)tan(π/4)):
We substitute the known values, tan(17π/12) = -1 and tan(π/4) = 1, into the expression:
(1 + (-1)(1)) = 0.
5. Since the denominator is zero, the expression is undefined. Thus, we cannot find its exact value.
Therefore, the correct answer is "undefined," not 1/sqrt(3).