Express 1+cos 30°/1-sin 30°
You MUST know the trig ratios of the 30-60-90 triangles by heart
sin 30° = 1/2, cos30° = √3/2
Assuming you meant
(1+cos 30°)/(1-sin 30°)
= (1 + √3/2) / (1 - 1/2)
= (1 + √3/2)(2)
= 2 + √3
or, using your half-angle formulas, that is cot(15°)
To simplify the expression 1+cos 30°/1-sin 30°, we can start by evaluating the trigonometric functions of 30°.
cos 30° = √3/2
sin 30° = 1/2
Substituting these values into the expression, we have:
1 + (√3/2) / 1 - (1/2)
Next, we simplify the numerator and denominator separately.
Numerator:
1 + (√3/2) = (2/2) + (√3/2) = (2 + √3)/2
Denominator:
1 - (1/2) = (2/2) - (1/2) = (2 - 1)/2 = 1/2
Substituting back into the original expression, we have:
(2 + √3)/2 / 1/2
Reciprocal of the denominator:
(2 + √3)/2 * 2/1
Simplifying the expression further:
(2 + √3)/1
Therefore, the simplified expression is 2 + √3.
To express the expression 1 + cos 30° / 1 - sin 30°, we can simplify it using trigonometric identities.
The identities we will use are:
1. Cosine double angle identity:
cos 2θ = 1 - 2sin²θ
2. Sine double angle identity:
sin 2θ = 2sinθcosθ
Now let's start:
cos 30° can be written as cos(2 * 15°) using the cosine double angle identity.
Applying the cosine double angle identity:
cos(2 * 15°) = 1 - 2 * sin²(15°)
Similarly, sin 30° can be written as sin(2 * 15°) using the sine double angle identity.
Applying the sine double angle identity:
sin(2 * 15°) = 2 * sin(15°) * cos(15°)
So, we have:
1 + cos 30° / 1 - sin 30° = 1 + (1 - 2sin²(15°)) / (1 - 2sin(15°)cos(15°))
Now, we need to find the values of sin(15°) and cos(15°).
To find these values, we can use the half-angle identities as follows:
sin(15°) = √[(1 - cos(30°))/2]
cos(15°) = √[(1 + cos(30°))/2]
Using the value of cos(30°) = √3 / 2, we can calculate these values:
sin(15°) = √[(1 - (√3 / 2))/2]
cos(15°) = √[(1 + (√3 / 2))/2]
Now we have all the values required to solve the expression:
1 + [(1 - 2sin²(15°)) / (1 - 2sin(15°)cos(15°))]
Substituting the values of sin(15°) and cos(15°), the expression becomes:
1 + [(1 - 2 * [(√(1 - (√3 / 2))/2)^2]) / (1 - 2 * √(1 - (√3 / 2))/2 * √(1 + (√3 / 2))/2)]
Simplifying it further using the values of sin(15°) and cos(15°), we can find the numerical value of the expression.