Trigonometry & Inverse Trigonometry
Trig Ratios Functions Identities
nta_abhyas_2025
Grade 11
Question:
Let $f(x) = \max \{\sin x, \cos x\}$. Then the number of roots of the equation $f(x) = \frac{1}{\sqrt{2}}$ in $(0, 2\pi)$ is
Step-by-Step Solution
Key Concept: The range of the maximum of two functions determines which horizontal lines can intersect the graph
From the graph of $\tan x$ and $\cot x$, we observe that the range of $f(x) = \max(\tan x, \cot x)$ is $(-\infty, -1] \cup [1, \infty)$. Since $g(x) = \frac{3}{4}$ is a constant function with value $\frac{3}{4}$, and this value lies in the interval $(-1, 1)$ which is not in the range of $f(x)$, the equation $f(x) = g(x)$ has no solutions.
Correct Answer: zero