Question 1

In the principle of mathematical induction, which of the following step is mandatory?

Accepted Answer

induction hypothesis

Question 2

What is the first step of mathematical induction?

Accepted Answer

Prove the first case, usually n = 1, is true .

Question 3

Which of the following correctly defines the sine of an angle in a right-angled triangle?

Accepted Answer

Ratio of the length of the side opposite the angle to the hypotenuse — Sine of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite to the angle to the hypotenuse. This is a fundamental trigonometric ratio used in geometry.

Question 4

An angle is measured as the length of the arc on a circle of radius 1 unit. What is the unit of this angle measure?

Accepted Answer

Radian — The radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. This is a natural unit of measuring angles in mathematics.

Question 5

Convert $\frac{3\pi}{4}$ radians into degrees.

Accepted Answer

135 degrees — Given: Radian measure = $\frac{3\pi}{4}$ radians
Find: Degree measure
Formula: Degree measure = $\frac{180}{\pi} 	imes$ Radian measure
Solution: Step 1: Substitute radian measure
$= \frac{180}{\pi} 	imes \frac{3\pi}{4}$
Step 2: Simplify
$= 180 	imes \frac{3}{4} = 135$
Answer: 135 degrees
Note: Remember to multiply by $\frac{180}{\pi}$ to convert radians to degrees.

Question 6

Describe the relationship between real numbers and radian measures of angles using the unit circle.

Accepted Answer

Every real number corresponds to a unique angle measured in radians on the unit circle. By wrapping the real number line around the unit circle, moving anticlockwise corresponds to positive real numbers and clockwise to negative real numbers. This establishes a one-to-one correspondence between real numbers and radian measures of angles. — The unit circle allows us to associate each real number with an angle measured in radians by considering the length of the arc from the initial point. Positive real numbers correspond to anticlockwise rotation, and negative real numbers correspond to clockwise rotation. This concept extends trigonometric functions to all real numbers.

Question 7

For a point $P(a,b)$ on the unit circle corresponding to angle $x$, what is the value of $\sin x$ and $\cos x$?

Accepted Answer

$\sin x = b$, $\cos x = a$ — On the unit circle, the point $P(a,b)$ has coordinates where $a$ is the x-coordinate and $b$ is the y-coordinate. By definition, $\cos x$ is the x-coordinate and $\sin x$ is the y-coordinate of the point corresponding to angle $x$.

Question 8

Identify the correct expression for $	an x$ in terms of coordinates $(a,b)$ of point $P$ on the unit circle.

Accepted Answer

$	an x = \frac{b}{a}$ — Tangent of angle $x$ is defined as the ratio of sine to cosine, i.e., $	an x = \frac{\sin x}{\cos x} = \frac{b}{a}$ where $a$ and $b$ are the x and y coordinates of the point on the unit circle.

Question 9

Explain why the trigonometric functions are periodic with period $2\pi$.

Accepted Answer

Trigonometric functions are periodic with period $2\pi$ because a complete revolution around the unit circle corresponds to an angle of $2\pi$ radians. After an angle increases by $2\pi$, the point on the unit circle repeats its position, causing the sine, cosine, and other trigonometric function values to repeat. — Since the unit circle has circumference $2\pi$, rotating by $2\pi$ radians returns the point to its original position. Therefore, the trigonometric functions, which depend on the coordinates of this point, repeat their values every $2\pi$ radians, establishing their periodicity.

Question 10

Given that $\sin 45^\circ = \frac{\sqrt{2}}{2}$, which quadrant does the angle $135^\circ$ lie in, and what is the sign of $\sin 135^\circ$?

Accepted Answer

Quadrant II, positive — An angle of $135^\circ$ lies in the second quadrant (between 90° and 180°). In the second quadrant, sine values are positive, so $\sin 135^\circ$ is positive.

Question 11

Using the unit circle, what is the value of $\cos ( - x )$ if $\cos x = a$?

Accepted Answer

a — Since cosine corresponds to the x-coordinate on the unit circle, and reflecting an angle $x$ to $-x$ changes the y-coordinate sign but not the x-coordinate, we have $\cos(-x) = \cos x = a$.

Question 12

Fill in the blank: The Pythagorean identity relating sine and cosine is _____ .

Accepted Answer

cos^2 x + sin^2 x = 1 / cos squared x plus sin squared x equals one — The Pythagorean identity states that for any angle $x$, the sum of the squares of sine and cosine is 1, i.e., $\cos^2 x + \sin^2 x = 1$. This follows from the equation of the unit circle $x^2 + y^2 = 1$.

Question 13

State and explain the identity relating $1 + 	an^2 x$ and $\sec^2 x$.

Accepted Answer

The identity is $1 + 	an^2 x = \sec^2 x$. It is derived from the Pythagorean identity by dividing both sides of $\cos^2 x + \sin^2 x = 1$ by $\cos^2 x$. This identity helps in simplifying expressions involving tangent and secant functions. — Starting from $\cos^2 x + \sin^2 x = 1$, divide both sides by $\cos^2 x$ to get $1 + 	an^2 x = \sec^2 x$ because $	an x = \frac{\sin x}{\cos x}$ and $\sec x = \frac{1}{\cos x}$. This identity is fundamental in trigonometric simplifications and solving equations.

Question 14

Which of the following is the domain of the function $	an x$?

Accepted Answer

All real numbers except $x 
eq (2n+1)\frac{\pi}{2}$, $n \in \mathbb{Z}$ — The tangent function is undefined where $\cos x = 0$, which occurs at $x = (2n+1)\frac{\pi}{2}$ for integer $n$. Hence, the domain excludes these points.

Question 15

What is the range of the function $\sec x$?

Accepted Answer

$(-\infty, -1] \cup [1, \infty)$ — Since $\sec x = \frac{1}{\cos x}$, and $\cos x$ lies between -1 and 1, $\sec x$ takes values less than or equal to -1 or greater than or equal to 1, but never between -1 and 1.

Trigonometric Functions

Trigonometric Functions — Study Notes

Introduction to Trigonometric Functions

Measuring Angles: Degrees and Radians

Relation between Radian Measure and Real Numbers

Practice Questions — Trigonometric Functions

All 14 Chapters in Mathematics