Use the sum, difference, double or half-angle formulas to find the exact value of 𝑡𝑎𝑛(75°).
tan(75°) = (tan60° + tan15°)/(1 - tan60° tan15°)
tan15° = (1-cos30°)/sin30°
now crank it out. You should get 2+√3
Since tan 15° requires extra calculations, try
tan(75) = tan(45 + 30)
= (1 + 1/√3) / (1 - (1)(1/√3)
= (1 + 1/√3) / (1 - 1/√3) * √3 / √3
= (√3 + 1)/(√3 - 1) * (√3 + 1)/(√3 + 1)
= (3 + 2√3 + 1)/2
= 2 + √3
well, duh - why did I not see that?
To find the exact value of 𝑡𝑎𝑛(75°) using the sum, difference, double, or half-angle formulas, we can use the following steps:
Step 1: Convert the given angle to radians.
To use trigonometric formulas, it's often easier to work with angles in radians. To convert from degrees to radians, we use the formula: radians = degrees × (π/180). So, for 75°, the angle in radians would be:
𝜃 = 75° × (𝜋/180) = (5𝜋/12) radians.
Step 2: Determine the reference angle.
The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. In this case, the reference angle for 75° is 75° - 180° = -105° (or 360° - 75° = 285° if measured counterclockwise from the positive x-axis). However, since we're working in radians, we'll consider the reference angle in radians, which would be:
𝜃_r = (-105°) × (𝜋/180) = (-7𝜋/12) radians.
Step 3: Use the half-angle formula for tangent (𝑡𝑎𝑛(𝜃/2)) to find 𝑡𝑎𝑛(𝜃).
The half-angle formula for tangent (tan) is given by:
𝑡𝑎𝑛(𝜃/2) = ± √((1 - 𝑐𝑜𝑠(𝜃)) / (1 + 𝑐𝑜𝑠(𝜃))).
In this case, we'll use the negative sign for the half-angle formula because the angle is in the second quadrant:
𝑡𝑎𝑛(75°) = -√((1 - 𝑐𝑜𝑠((-7𝜋/12))) / (1 + 𝑐𝑜𝑠((-7𝜋/12)))).
Note: In this step, it's important to have the values of cosine (cos) and sine (sin) of the reference angle, which can be found in reference charts or calculated using trigonometric ratios or special triangles.
Step 4: Simplify the expression.
To simplify the expression, substitute the values of cosine and sine obtained from step 3 into the formula and calculate the result.
𝑡𝑎𝑛(75°) = -√((1 - 𝑐𝑜𝑠((-7𝜋/12))) / (1 + 𝑐𝑜𝑠((-7𝜋/12)))).
Step 5: Calculate the value.
Using a scientific calculator or computer software, you can substitute the value of 𝑐𝑜𝑠((-7𝜋/12)) and 𝑠𝑖𝑛((-7𝜋/12)) into the formula and calculate the result. The calculated value will give you the exact value of 𝑡𝑎𝑛(75°).
Note: The final value may be an exact fraction or a decimal approximation depending on the given angle and the accuracy of the calculator or software used.
By following these steps, you can use the sum, difference, double, or half-angle formulas to find the exact value of 𝑡𝑎𝑛(75°).