Full Grade 11 test

The period of the tangent function is 180°. Therefore, adding or subtracting any integer multiple of 180° does not change the tangent value.

Rule: tan(α + 180° · k) = tan α, where k is an integer.

1. Notice that 180° · k has been added to the angle.
2. This is a whole number of tangent periods, so that part may be removed.
3. The expression becomes tan α.

Thus, keep the tangent function and the original angle that appears before the added multiple of 180°.

A root taken from another root can be written as one root.

1. Multiply the indices of the outer and inner roots.
2. Take the root with that new index from the original number.
3. Check the answer: raising it to the product of the two indices must give the original number.

Rule: the nth root of the mth root of a number is the (n × m)th root of that number.

Determine the domain conditions for both logarithms. Since the base is between 0 and 1, the logarithmic function is decreasing, so reverse the inequality when comparing the arguments. Intersect the solution of the linear inequality with the domain of both arguments.

The expression a sin x + b cos x can be written as R sin(x + φ), where R = √(a² + b²). Since the maximum value of sine is 1, the maximum value of the whole expression is √(a² + b²).