Competitive Exams / CUET UG / Trigonometry & Co-ordinate Geometry

Trigonometry & Co-ordinate Geometry for CUET UG

Master trigonometric ratios, identities, standard angles, heights and distances, distance and midpoint formulas, section formula, centroid, slope, and collinearity with diagram-led notes, solved examples, and timed practice.

Open Practice Jump to Notes

Ad Slot: Top Navigation Leaderboard

Overview

Why This Combined Chapter Matters in CUET

Trigonometry rewards value recall, identity discipline, and correct triangle interpretation. Co-ordinate geometry rewards formula accuracy, clean substitution, and geometric reasoning through algebra.

Put together, they create a high-value CUET block because many questions are direct once the right formula or theorem is recognized in the first few seconds.

Ad Slot: High CTR Rectangle after Overview

Section A

Notes & Visual Concept Builder

Reduce the problem to a right triangle or a formula pattern

Diagram 1 - Right Triangle and Ratios

Angle theta is at A. The opposite side is perpendicular, the adjacent side is base, and the longest side is the hypotenuse.

Diagram 2 - Standard Angle Memory Strip

Ratio	0°	30°	45°	60°	90°
sin	0	1/2	1/√2	√3/2	1
cos	1	√3/2	1/√2	1/2	0
tan	0	1/√3	1	√3	∞

Sin rises from 0 to 1, cos falls from 1 to 0, and tan grows from 0 to infinity as the angle moves from 0° to 90°.

Diagram 3 - Quadrants and Signs

Quadrant I

All ratios positive

Quadrant II

sin, cosec positive

Quadrant III

tan, cot positive

Quadrant IV

cos, sec positive

ASTC rule: All positive in Quadrant I, Sine positive in Quadrant II, Tangent positive in Quadrant III, and Cosine positive in Quadrant IV.

Diagram 4 - Heights and Distances Setup

The line of sight becomes the hypotenuse. Tower or building height is vertical, and the ground distance is horizontal.

Diagram 5 - Distance and Midpoint

Distance uses a Pythagoras pattern on the coordinate plane; midpoint simply averages the coordinates.

1. The Six Trigonometric Ratios 2. Core Identities and Standard Angles 3. Quadrant Signs and Periodicity 4. Heights, Distances, Elevation, and Depression 5. Distance Formula and Midpoint Formula 6. Section Formula and Centroid 7. Slope, Parallel Lines, and Perpendicular Lines 8. Collinearity and Area of Triangle

1. The Six Trigonometric Ratios

In a right triangle, trigonometric ratios connect an acute angle with the three sides of the triangle. For CUET, the main task is to identify the opposite side, adjacent side, and hypotenuse correctly before applying a ratio.

\sin\theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}}, \quad \cos\theta = \frac{\text{Base}}{\text{Hypotenuse}}, \quad \tan\theta = \frac{\text{Perpendicular}}{\text{Base}}

\cot\theta = \frac{1}{\tan\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \cosec\theta = \frac{1}{\sin\theta}

Once the angle is fixed, the naming of base and perpendicular becomes angle-dependent. This is why students sometimes get the correct formula but wrong substitution.

2. Core Identities and Standard Angles

The three big identities are enough to solve a large fraction of direct trigonometry questions in CUET. Combine them with the standard values of 0°, 30°, 45°, 60°, and 90° and many problems collapse instantly.

\sin^2\theta + \cos^2\theta = 1

1 + \tan^2\theta = \sec^2\theta \qquad 1 + \cot^2\theta = \cosec^2\theta

10-second shortcut: Memorize sin values in order as

0, \frac12, \frac{1}{\sqrt2}, \frac{\sqrt3}{2}, 1

. Cos is the same list in reverse.

3. Quadrant Signs and Periodicity

When angles exceed 90° or 360°, reduce them to a reference angle and determine the sign from the quadrant. The ASTC rule keeps this fast: All, Sine, Tangent, Cosine.

\tan(180^\circ n + \theta) = \tan\theta, \qquad \tan(180^\circ n - \theta) = -\tan\theta

\sin(360^\circ n + \theta) = \sin\theta, \qquad \cos(360^\circ n + \theta) = \cos\theta

Always reduce the angle first and decide the sign after locating the final quadrant.

4. Heights, Distances, Elevation, and Depression

Heights and distances convert real-world viewing situations into right triangles. The two key words are angle of elevation and angle of depression. A depression angle at the top equals the elevation angle at the ground because of alternate interior angles.

\tan\theta = \frac{\text{height}}{\text{horizontal distance}}

If the thread of a kite, a ladder, or the line of sight is given, that often becomes the hypotenuse. If a building or tree stands vertically, that becomes the perpendicular.

Exam trap: Do not treat the angle of depression and angle of elevation as different triangle angles. They are equal when the horizontal lines are parallel.

5. Distance Formula and Midpoint Formula

Coordinate geometry begins with location on the Cartesian plane. The distance formula is just the Pythagoras theorem written for two points, and the midpoint formula is coordinate averaging.

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)

CUET often hides these formulas inside words such as equidistant, midpoint, distance from origin, or equal lengths.

6. Section Formula and Centroid

If a point divides a line segment internally in the ratio $m:n$ , the coordinates of the dividing point are weighted averages of the endpoints. The midpoint formula is just the special case where $m=n=1$ .

P = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)

G = \left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)

The centroid formula is extremely useful when two vertices are known and the centroid is given. You can recover the third vertex directly.

7. Slope, Parallel Lines, and Perpendicular Lines

Slope measures steepness. A positive slope rises from left to right, a negative slope falls, a horizontal line has slope 0, and a vertical line has undefined slope.

m = \frac{y_2-y_1}{x_2-x_1}

m_1 = m_2 \text{ for parallel lines} \qquad \text{and} \qquad m_1m_2 = -1 \text{ for perpendicular lines}

Whenever a question mentions perpendicular, parallel, or inclination, slope is usually the fastest route.

8. Collinearity and Area of Triangle

Three points are collinear if they lie on the same straight line. A powerful test is that the area of the triangle formed by the three points becomes zero.

\text{Area} = \frac12\left|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)\right|

10-second shortcut: If the question explicitly says the points are collinear, set the area expression to zero and solve for the unknown directly.

Solved Practice

Solved Examples

Try first, then open the reasoning

Exam Trap: Most errors here come from choosing the wrong side in trigonometry, forgetting periodicity signs, or mixing up midpoint and section formula.

Example 1: Find $\dfrac{\sin 45^\circ \cdot \cos 60^\circ}{\tan 30^\circ}$.

Substitute the standard values:

\frac{\frac{1}{\sqrt2}\cdot\frac12}{\frac{1}{\sqrt3}} = \frac{1}{2\sqrt2}\cdot\sqrt3 = \frac{\sqrt3}{2\sqrt2}

Example 2: Find $4\cos^2 60^\circ - \sin^4 30^\circ$.

4\left(\frac12\right)^2 - \left(\frac12\right)^4 = 4\cdot\frac14 - \frac1{16} = 1 - \frac1{16} = \frac{15}{16}

Example 3: If $\sin\theta = \frac58$, find $\cos\theta$.

Take perpendicular $=5$ and hypotenuse $=8$ .

Base $=\sqrt{8^2-5^2}=\sqrt{39}$ .

\cos\theta = \frac{\sqrt{39}}{8}

Example 4: Find $\tan 860^\circ$.

860^\circ = 180^\circ \times 5 - 40^\circ

Using periodicity,

\tan(180n-\theta) = -\tan\theta

\tan 860^\circ = -\tan 40^\circ

Example 5: Which quadrant contains $485^\circ$?

Subtract $360^\circ$ once:

485^\circ - 360^\circ = 125^\circ

$125^\circ$ lies between $90^\circ$ and $180^\circ$ , so it lies in Quadrant II.

Example 6: From the top of a 100 m tower, the angle of depression of a car is $30^\circ$. Find the horizontal distance of the car.

The angle of depression equals the angle of elevation.

Let the distance be $x$ .

\tan30^\circ = \frac{100}{x} = \frac{1}{\sqrt3}

x = 100\sqrt3 \text{ m}

Example 7: A kite flies with a 150 m string making $30^\circ$ with the horizontal. Find the height of the kite.

The string is the hypotenuse.

Height

=150\sin30^\circ = 150\cdot\frac12 = 75 \text{ m}

Example 8: From 20 m away from the foot of a tower, the angle of elevation is $30^\circ$. Find the height of the tower.

Let the height be $h$ .

\tan30^\circ = \frac{h}{20} = \frac1{\sqrt3}

h = \frac{20}{\sqrt3} \text{ m}

Example 9: Find the distance between $(x,6)$ and $(3,0)$ if it is 10 units. Find $x$.

Use the distance formula:

\sqrt{(x-3)^2 + (6-0)^2} = 10

(x-3)^2 + 36 = 100 \Rightarrow (x-3)^2 = 64

x = 11 \text{ or } -5

Example 10: Find the ratio in which the x-axis divides the segment joining $(4,-6)$ and $(1,3)$.

Let the ratio be $k:1$ .

The point on the x-axis has y-coordinate 0, so

\frac{k\cdot3 + 1\cdot(-6)}{k+1} = 0

Thus

3k-6=0 \Rightarrow k=2

The ratio is

2:1

Example 11: Two vertices of a triangle are $(-2,5)$ and $(-4,4)$ and its centroid is the origin. Find the third vertex.

Let the third vertex be $(x,y)$ .

\frac{-2-4+x}{3}=0 \Rightarrow x=6

\frac{5+4+y}{3}=0 \Rightarrow y=-9

So the third vertex is $(6,-9)$ .

Example 12: Find the slope of the line perpendicular to the line through $(2,6)$ and $(-3,1)$.

Slope of the given line:

m_1 = \frac{6-1}{2-(-3)} = \frac55 = 1

For a perpendicular line,

m_2 = -\frac{1}{m_1} = -1

Ad Slot: Inline Feed between Notes and Practice

Section B

The Test Zone

This chapter has a separate practice route with 4 sectional sessions of 10 questions each and a mixed 40-question mock. Every question runs on a 60-second timer so we train value recall, visual setup, and formula execution together.

Session 1

Ratios, Identities & Standard Angles

Core trig values, reciprocal relations, and identity-based simplifications.

Session 2

Quadrants & Heights

Elevation, depression, periodicity, and applied right-triangle setups.

Session 3

Distance, Midpoint & Section Formula

Point geometry, centroid, ratio division, and equidistance patterns.

Session 4

Slope, Collinearity & Mixed Revision

Slope, perpendicular lines, collinearity, and mixed concept combinations.

Start Practice

Finished this topic?

Keep the practice loop moving

Move straight from chapter-wise questions into a subject test, then loop back into weaker areas instead of ending the session here.

Take Quantitative Aptitude Mock Test Try Full CUET Mock