Trigonometric, exponential and logarithmic functions
Trigonometric, exponential, and logarithmic functions are examined through equations, inequalities, graph transformations, and rates of change. The key is to respect domains and periods before manipulating expressions.
Core rules
- Know the standard identities, especially sin^2 x+cos^2 x=1, double-angle formulas, and tan x=sin x/cos x. Work in radians unless degrees are stated.
- For sin θ=k or cos θ=k, find all solutions in the required interval using symmetry and periodicity; do not divide by a factor that may be zero.
- exp and log are inverse functions: log_a(xy)=log_a x+log_a y, log_a(x^r)=r log_a x, and a^{log_a x}=x. The domain of log is x>0.
- Graph transformations: y=f(x-h)+k shifts; y=af(bx) scales. Asymptotes for exponentials and logs must be stated.
- Related rates use the chain rule: if y depends on x and x depends on t, then dy/dt=(dy/dx)(dx/dt).
Worked example
Solve 2^{x+1}=8^{x-1}. Since 8=2^3, the right side is 2^{3(x-1)}. Equal bases give x+1=3x-3, hence 4=2x and x=2. Check: left side 2^3=8, right side 8^1=8. For a related-rate variant, if y=e^{2x}, then dy/dx=2e^{2x}; when x=1 and dx/dt=3, dy/dt=2e^2·3=6e^2.
Traps
Squaring can introduce extraneous roots, and logarithms cannot accept zero or negative inputs. When solving trig equations, dividing by cos x or sin x can lose whole families of solutions. In inequalities, reversing a log or exponential inequality requires attention to whether the base is greater than 1 or between 0 and 1. In related rates, differentiate before substituting numerical values.