Write each expression as sine, cosine, or tangent of an angle. Then find the exact value of the expression
tan 50-tan20/1+ tan 50 tan20
since tan(A-B) = (tanA-tanB)/(1-tanAtanB) I'd say you're looking at tan30 = 1/√3
To write the expression as sine, cosine, or tangent of an angle, we need to simplify the expression using trigonometric identities.
First, let's simplify the expression by factoring out the common factor of tan(50) in the numerator and tan(20) in the denominator:
(tan(50) - tan(20)) / (1 + tan(50) * tan(20))
Using the identity tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A) * tan(B)), we can rewrite the expression as:
tan(50 - 20) / (1 + tan(50) * tan(20))
This simplifies to:
tan(30) / (1 + tan(50) * tan(20))
Now, using the values of sin, cos, and tan of common angles, we know that tan(30) = sin(30) / cos(30).
sin(30) = 1/2, and cos(30) = √3/2.
So, tan(30) = (1/2) / (√3/2) = 1/√3 = √3/3.
Now, substituting the value of tan(30) into the expression:
(√3/3) / (1 + tan(50) * tan(20))
We need to find the values of tan(50) and tan(20) using a calculator or trigonometric tables.
Using a calculator, we find that tan(50) ≈ 1.1918, and tan(20) ≈ 0.3639.
Substituting these values into the expression:
(√3/3) / (1 + (1.1918)(0.3639))
Calculating the denominator:
(√3/3) / (1 + 0.4346)
Simplifying further:
(√3/3) / 1.4346
Multiplying both numerator and denominator by 3 to get rid of the fraction:
√3 / (3 * 1.4346)
√3 / 4.3038
Finally, simplifying:
√3 / 4.3038 ≈ 0.4430
Therefore, the exact value of the expression is approximately 0.4430.