krit.club logo

Trigonometric Functions - Relation between radian and real numbers

Grade 11CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

The relationship between real numbers and radian measures is established using a unit circle (radius r=1r = 1) centered at the origin (0,0)(0,0). For any real number xx, we can associate it with a point PP on the unit circle such that the length of the arc APAP is exactly ∣x∣|x| units, where AA is the point (1,0)(1,0). This demonstrates that a real number xx can be viewed as the radian measure of the angle it subtends at the center.

A unit circle showing the correspondence between an arc length x and a point P on the circumference.
•

Consider a vertical number line (the real line) tangent to the unit circle at point A(1,0)A(1,0). If we 'wrap' this number line around the circle, every real number xx on the line coincides with a unique point on the circle. Positive real numbers wrap in the counter-clockwise direction, while negative real numbers wrap in the clockwise direction.

A vertical real number line tangent to a unit circle at (1,0) illustrating the wrapping process.
•

Due to this wrapping property, the set of real numbers and radian measures are essentially the same. For any real number xx, the angle subtended at the center of the unit circle has a measure of xx radians. Therefore, 'radian measure' and 'real number' can be used interchangeably in trigonometric contexts.

Diagram showing that the arc length of a unit circle is equal to the angle in radians.
•

In a unit circle, the circumference is 2π2\pi. This means one complete revolution corresponds to the real number 2π2\pi (approx 6.286.28). Consequently, any real number xx can be reduced to a value within [0,2π)[0, 2\pi) to find its position on the circle by taking x(mod2π)x \pmod{2\pi}.

📐Formulae

θ=lr\theta = \frac{l}{r}

In a unit circle: θ(radians)=l(arc length)\text{In a unit circle: } \theta (\text{radians}) = l (\text{arc length})

π radians=180∘\pi \text{ radians} = 180^\circ

1 radian=180∘π≈57∘16′1 \text{ radian} = \frac{180^\circ}{\pi} \approx 57^\circ 16'

💡Examples

Problem 1:

Given a unit circle, if a real number x=3π2x = \frac{3\pi}{2} is mapped onto the circle starting from (1,0)(1,0) in the counter-clockwise direction, find the radian measure of the angle subtended at the center.

Solution:

In a unit circle, the radius r=1r = 1. The real number xx corresponds to the arc length ll. Therefore, l=3π2l = \frac{3\pi}{2}. Using the formula θ=lr\theta = \frac{l}{r}, we get θ=3π/21=3π2\theta = \frac{3\pi/2}{1} = \frac{3\pi}{2} radians.

Explanation:

Because the radius is 1, the numerical value of the real number on the number line is identical to the radian measure of the angle it subtends at the center.

Problem 2:

If the real number x=−2πx = -2\pi is wrapped around a unit circle, what is the final position on the coordinate plane?

Solution:

The length of the arc is ∣−2π∣=2π|-2\pi| = 2\pi. Since the circumference of a unit circle is C=2πr=2π(1)=2πC = 2\pi r = 2\pi(1) = 2\pi, a distance of 2π2\pi represents one full revolution. The negative sign indicates a clockwise direction. Starting from (1,0)(1,0) and moving clockwise for 2π2\pi brings us back to (1,0)(1,0).

Explanation:

The real number −2π-2\pi corresponds to an angle of −2π-2\pi radians, which is a complete rotation, returning the point to the starting position on the x-axis.

Problem 3:

Determine the coordinates of the point on the unit circle corresponding to the real number x=5π6x = \frac{5\pi}{6}.

Unit circle with a radius at 150 degrees representing the real number 5pi/6.

Solution:

  1. In a unit circle, the real number xx corresponds to an angle θ=x\theta = x radians.
  2. Here, θ=5π6\theta = \frac{5\pi}{6} radians.
  3. Convert to degrees: 5π6×180π=150∘\frac{5\pi}{6} \times \frac{180}{\pi} = 150^\circ.
  4. The coordinates of a point PP on the unit circle are (cos⁡θ,sin⁡θ)(\cos \theta, \sin \theta).
  5. xx-coordinate = cos⁡(150∘)=cos⁡(180∘−30∘)=−cos⁡30∘=−32\cos(150^\circ) = \cos(180^\circ - 30^\circ) = -\cos 30^\circ = -\frac{\sqrt{3}}{2}.
  6. yy-coordinate = sin⁡(150∘)=sin⁡(180∘−30∘)=sin⁡30∘=12\sin(150^\circ) = \sin(180^\circ - 30^\circ) = \sin 30^\circ = \frac{1}{2}.
  7. Coordinates are (−32,12)\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right).

Explanation:

Since the radius is 1, the arc length 5π6\frac{5\pi}{6} is equivalent to the central angle in radians. We use the standard trigonometric ratios for 150∘150^\circ to find the point on the coordinate plane.

Problem 4:

If a real number x=π4x = \frac{\pi}{4} is represented on the unit circle, find the length of the chord joining the starting point A(1,0)A(1,0) and the point PP corresponding to xx.

A unit circle with a chord connecting (1,0) to the point corresponding to pi/4.

Solution:

  1. The real number x=π4x = \frac{\pi}{4} corresponds to an angle of θ=45∘\theta = 45^\circ at the center.
  2. Point AA is (1,0)(1,0).
  3. Point PP is (cos⁡45∘,sin⁡45∘)=(12,12)(\cos 45^\circ, \sin 45^\circ) = (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}).
  4. Distance d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
  5. d=(12−1)2+(12−0)2d = \sqrt{(\frac{1}{\sqrt{2}} - 1)^2 + (\frac{1}{\sqrt{2}} - 0)^2}
  6. d=(12+1−2)+12=2−2d = \sqrt{(\frac{1}{2} + 1 - \sqrt{2}) + \frac{1}{2}} = \sqrt{2 - \sqrt{2}}.

Explanation:

We first identify the position of the real number on the circle as a coordinate point and then use the distance formula to find the chord length.