Find the exact value of the following expression \
\ using a sum/difference formula
The Correct Answer and Explanation is:
The exact value of the expression is on its way!
Now, let’s break it down using a sum or difference identity. If the expression involves trigonometric functions like sin(75°), cos(15°), or tan(105°), we can apply the appropriate identity:
For example, to find sin(75°), we use the identity: sin(A + B) = sin A cos B + cos A sin B
Let A = 45° and B = 30°, so: sin(75°) = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4
This method uses known values of sine and cosine for standard angles. These identities are powerful tools in trigonometry because they allow us to evaluate non-standard angles by expressing them as sums or differences of angles with known trigonometric values.
The same approach works for cosine and tangent: cos(A − B) = cos A cos B + sin A sin B tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
These formulas are derived from the unit circle and the definitions of sine, cosine, and tangent in terms of coordinates. They are essential in simplifying expressions, solving equations, and proving identities.
Using these identities not only helps in exact evaluations but also deepens understanding of how trigonometric functions behave under transformations. It reinforces the interconnectedness of angles and their trigonometric values, which is foundational in both pure and applied mathematics.
