Trigonometric Functions Ex 3.1: Understanding Angles and Radians

Understanding Angles: Degrees and Radians

Angle

An angle is a measure of the rotation of a given ray about its initial point. The initial position of the ray is called the initial side, and the final position after rotation is called the terminal side. The point of rotation is the vertex. Positive angles are generated by anticlockwise rotation, and negative angles by clockwise rotation.

Degree Measure

If the rotation from the initial to terminal side is (1/360)th of a revolution, the angle is said to have a measure of one degree, denoted as 1°. A degree is further subdivided into 60 minutes (60') and a minute into 60 seconds (60''). So, 1° = 60' and 1' = 60''.

Radian Measure

The angle subtended at the centre by an arc of a circle whose length is equal to the radius of the circle is called one radian, denoted as 1c or simply 1. It's a unit of angle measurement based on the ratio of arc length to radius.

Converting Between Degree and Radian Measures

The ability to convert angles between degree and radian measures is absolutely critical in trigonometry. While degrees are intuitive for visualising rotations (like 360° for a full circle), radians are more natural in many mathematical formulas, especially those involving calculus, as they are a ratio and thus dimensionless.

Consider a circle with radius 'r'. The circumference of the circle is $2\pi r$. The angle subtended by the entire circumference at the centre is 360°. By definition, an arc of length 'r' subtends an angle of 1 radian. Therefore, an arc of length $2\pi r$ (the full circumference) subtends an angle of $2\pi$ radians.

This gives us the fundamental relationship: $2\pi \text{ radians} = 360\degree$.

Simplifying this, we get: $\pi \text{ radians} = 180\degree$.

From this, we can derive the conversion factors:

1. To convert degrees to radians: Multiply the degree measure by $\frac{\pi}{180\degree}$. For example, $60\degree = 60 \times \frac{\pi}{180} \text{ radians} = \frac{\pi}{3} \text{ radians}$.
2. To convert radians to degrees: Multiply the radian measure by $\frac{180\degree}{\pi}$. For example, $\frac{\pi}{2} \text{ radians} = \frac{\pi}{2} \times \frac{180\degree}{\pi} = 90\degree$.

It's important to remember that when an angle is expressed without any unit, it is generally understood to be in radians. For instance, '$\pi

Arc Length and Angle Relation: Worked Examples

• Example 1: Convert 40 degrees 20 minutes to radian measure. Step 1: Convert minutes to degrees. We know that 60 minutes = 1 degree. So, 20 minutes = $20/60 = 1/3$ degree. Step 2: Add this to the degree part. Total degrees = $40 + 1/3 = 121/3$ degrees. Step 3: Use the conversion factor from degrees to radians. Multiply by $\frac{\pi}{180}$. Radian measure = $\frac{121}{3} \times \frac{\pi}{180} = \frac{121\pi}{540}$ radians. Final Answer: $40\degree 20' = \frac{121\pi}{540}$ radians.
• Example 2: Convert 6 radians to degree measure (Use $\pi = 22/7$). Step 1: Use the conversion factor from radians to degrees. Multiply by $\frac{180}{\pi}$. Degree measure = $6 \times \frac{180}{\pi}$ degrees. Step 2: Substitute the value of $\pi = 22/7$. Degree measure = $6 \times \frac{180}{22/7} = 6 \times \frac{180 \times 7}{22} = 6 \times \frac{90 \times 7}{11} = \frac{540 \times 7}{11} = \frac{3780}{11}$ degrees. Step 3: Convert the improper fraction to mixed number and then to degrees, minutes, and seconds. $\frac{3780}{11} = 343 \frac{7}{11}$ degrees. So, 343 degrees. Step 4: Convert the fractional part of a degree to minutes. Multiply by 60. $\frac{7}{11} \times 60 = \frac{420}{11} = 38 \frac{2}{11}$ minutes. So, 38 minutes. Step 5: Convert the fractional part of a minute to seconds. Multiply by 60. $\frac{2}{11} \times 60 = \frac{120}{11} \approx 10.9$ seconds. Approximately 11 seconds. Final Answer: 6 radians $\approx 343\degree 38' 11''$.
• Example 3: Find the radius of the circle in which a central angle of $60\degree$ intercepts an arc of length 37.4 cm (Use $\pi = 22/7$). Step 1: The formula for arc length is $l = r\theta$, where $\theta$ must be in radians. Step 2: Convert the given angle from degrees to radians. $\theta = 60\degree = 60 \times \frac{\pi}{180} = \frac{\pi}{3}$ radians. Step 3: Given arc length $l = 37.4$ cm. Step 4: Substitute the values into the formula $l = r\theta$. $37.4 = r \times \frac{\pi}{3}$ Step 5: Solve for r. $r = \frac{37.4 \times 3}{\pi}$ Step 6: Substitute $\pi = 22/7$. $r = \frac{37.4 \times 3}{22/7} = \frac{37.4 \times 3 \times 7}{22} = \frac{112.2 \times 7}{22} = \frac{785.4}{22} = 35.7$ cm. Final Answer: The radius of the circle is 35.7 cm.

Exam Tip: Avoiding Common Mistakes

Students often make a few recurring mistakes when dealing with angle measures and arc length problems. Here are some tips to avoid them:

1. Always check the units! This is the most crucial step. The formula $l = r\theta$ is ONLY valid when $\theta$ is in radians. If the angle is given in degrees, your first step should always be to convert it to radians. Forgetting this will lead to incorrect answers.
2. Precision with $\pi$: Unless specified, use the exact value of $\pi$ in your calculations or leave answers in terms of $\pi$. If an approximation like $22/7$ or $3.14$ is given, use that specific value. Otherwise, your answer might differ from the marking scheme.
3. Degree to minute/second conversion: Remember the factors: $1\degree = 60'$ and $1' = 60''$. When converting degrees with decimal parts to degrees, minutes, and seconds, multiply the fractional part by 60 for minutes, and the remaining fractional part of minutes by 60 for seconds. Don't round off too early.
4. Positive vs. Negative Angles: Be mindful of the direction of rotation. Anticlockwise is positive, clockwise is negative. While Ex 3.1 primarily deals with magnitude, understanding this convention is fundamental.

Practice Questions with Solutions

• Q: Find the radian measure corresponding to $-47\degree 30'$. A: Step 1: Convert minutes to degrees. $30' = 30/60 = 0.5\degree$. Step 2: Add this to the degree part. Total degrees = $- (47 + 0.5)\degree = -47.5\degree$. Step 3: Convert degrees to radians using the factor $\frac{\pi}{180}$. Radian measure = $-47.5 \times \frac{\pi}{180} = - \frac{475}{10} \times \frac{\pi}{180} = - \frac{95}{2} \times \frac{\pi}{180} = - \frac{19}{2} \times \frac{\pi}{36} = - \frac{19\pi}{72}$ radians. Final answer: $-47\degree 30' = - \frac{19\pi}{72}$ radians.
• Q: Find the degree measure corresponding to $\frac{11}{16}$ radians. (Use $\pi = 22/7$) A: Step 1: Use the conversion factor from radians to degrees, which is $\frac{180}{\pi}$. Degree measure = $\frac{11}{16} \times \frac{180}{\pi}$. Step 2: Substitute $\pi = 22/7$. Degree measure = $\frac{11}{16} \times \frac{180}{22/7} = \frac{11}{16} \times \frac{180 \times 7}{22}$. Step 3: Simplify the expression. Degree measure = $\frac{11}{16} \times \frac{90 \times 7}{11} = \frac{1}{16} \times 90 \times 7 = \frac{630}{16} = \frac{315}{8}$ degrees. Step 4: Convert to degrees, minutes, and seconds. $\frac{315}{8} = 39 \frac{3}{8}$ degrees. So, 39 degrees. Step 5: Convert fractional part of degree to minutes. $\frac{3}{8} \times 60 = \frac{180}{8} = \frac{45}{2} = 22 \frac{1}{2}$ minutes. So, 22 minutes. Step 6: Convert fractional part of minute to seconds. $\frac{1}{2} \times 60 = 30$ seconds. Final answer: $\frac{11}{16}$ radians = $39\degree 22' 30''$.
• Q: In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of the minor arc of the chord. A: Step 1: The diameter is 40 cm, so the radius $r = 40/2 = 20$ cm. Step 2: The length of the chord is also 20 cm. Since the radius is 20 cm and the chord length is 20 cm, the triangle formed by the two radii and the chord is an equilateral triangle. Step 3: In an equilateral triangle, all angles are $60\degree$. So, the central angle $\theta = 60\degree$. Step 4: Convert the central angle to radians. $\theta = 60 \times \frac{\pi}{180} = \frac{\pi}{3}$ radians. Step 5: Use the arc length formula $l = r\theta$. $l = 20 \times \frac{\pi}{3} = \frac{20\pi}{3}$ cm. Final answer: The length of the minor arc is $\frac{20\pi}{3}$ cm.
• Q: If in two circles, arcs of the same length subtend angles $60\degree$ and $75\degree$ at the centre, find the ratio of their radii. A: Step 1: Let the radii of the two circles be $r_1$ and $r_2$, and the common arc length be $l$. Let the angles be $\theta_1 = 60\degree$ and $\theta_2 = 75\degree$. Step 2: Convert angles to radians. $\theta_1 = 60 \times \frac{\pi}{180} = \frac{\pi}{3}$ radians. $\theta_2 = 75 \times \frac{\pi}{180} = \frac{5\pi}{12}$ radians. Step 3: Use the arc length formula $l = r\theta$ for both circles. For the first circle: $l = r_1 \theta_1 \implies l = r_1 \frac{\pi}{3}$. So, $r_1 = \frac{3l}{\pi}$. For the second circle: $l = r_2 \theta_2 \implies l = r_2 \frac{5\pi}{12}$. So, $r_2 = \frac{12l}{5\pi}$. Step 4: Find the ratio $r_1 : r_2$. $\frac{r_1}{r_2} = \frac{3l/\pi}{12l/5\pi} = \frac{3l}{\pi} \times \frac{5\pi}{12l} = \frac{3 \times 5}{12} = \frac{15}{12} = \frac{5}{4}$. Final answer: The ratio of their radii is $5:4$.

Frequently Asked Questions