Without using a calculator compare the values of tan60 and tan70. Explain your reasoning.
since tan(x) is increasing on 0 < x < 90, tan60 < tan70
To compare the values of tan(60°) and tan(70°) without using a calculator, we need to rely on some trigonometric identities and understand the values of tangent for some common angles.
First, let's recall the definition of the tangent function:
tan(x) = sin(x) / cos(x)
Now, let's consider some commonly known trigonometric values:
sin(30°) = 1/2 and cos(30°) = √3/2
sin(45°) = √2/2 and cos(45°) = √2/2
sin(60°) = √3/2 and cos(60°) = 1/2
We can use these known values to find the values of tan(60°) and tan(70°) indirectly.
For tan(60°), we substitute the values of sin(60°) and cos(60°) into the tangent formula:
tan(60°) = (sqrt(3)/2) / (1/2) = sqrt(3)
For tan(70°), we do not have a known value for sin(70°) or cos(70°). However, we can use the identity tan(x) = 1 / cot(x) and cot(x) = cos(x) / sin(x) to find the value of tan(70°).
cot(20°) = cos(20°) / sin(20°)
We can look up the values of sin(20°) and cos(20°), which are approximately 0.342 and 0.940, respectively.
cot(20°) = 0.940 / 0.342 ≈ 2.748
Now, the inverse of the cotangent is the tangent:
tan(70°) ≈ 1 / cot(20°) ≈ 1 / 2.748 ≈ 0.363
Therefore, the approximate values of tan(60°) and tan(70°) are sqrt(3) and 0.363, respectively.
Comparing the two values, we can conclude that tan(60°) is greater than tan(70°), as sqrt(3) is approximately 1.732, which is larger than 0.363.