Class 10 Maths · 2026–27 study draft

Introduction to Trigonometry: Ratios That Connect Sides

Trigonometry connects an angle in a right-angled triangle with ratios of its sides. Similar right triangles with the same acute angle have the same side ratios, even when their sizes differ. Identify the reference angle first; only then label opposite and adjacent sides.

What you will learn

  • Label opposite, adjacent and hypotenuse relative to an angle.
  • Calculate all six ratios from side lengths.
  • Use exact values for standard angles.
  • Apply simple identities while respecting where ratios are defined.

The CBSE 2026–27 syllabus covers ratios for acute angles of right triangles, values at 30°, 45° and 60°, defined ratios at 0° and 90°, relationships between ratios, and simple identities based on sine squared plus cosine squared equals one. Heights and distances are a related, separate topic.

Core concepts

The angle controls the side labels

The hypotenuse is opposite the right angle and is the longest side. For an acute angle θ\theta, the opposite side faces θ\theta and the adjacent side touches θ\theta but is not the hypotenuse. Opposite and adjacent swap when you choose the other acute angle.

Ratios have no units

Let OO, AA and HH denote opposite, adjacent and hypotenuse. Sine is O/HO/H, cosine is A/HA/H and tangent is O/AO/A. Use the same unit for every side. Secant, cosecant and cotangent are the reciprocals of cosine, sine and tangent respectively, wherever the denominator is non-zero.

Use exact standard-angle values

At 30∘30^\circ, sine is 1/21/2, cosine is 3/2\sqrt3/2 and tangent is 1/31/\sqrt3. At 45∘45^\circ, sine and cosine are 1/21/\sqrt2 and tangent is 11. At 60∘60^\circ, sine is 3/2\sqrt3/2, cosine is 1/21/2 and tangent is 3\sqrt3. Keep surds rather than rounding early.

Endpoints need care

The endpoint values are sin⁡0∘=0\sin0^\circ=0, cos⁡0∘=1\cos0^\circ=1, sin⁡90∘=1\sin90^\circ=1 and cos⁡90∘=0\cos90^\circ=0. Tangent and secant at 90∘90^\circ, and cotangent and cosecant at 0∘0^\circ, are undefined because their definitions would divide by zero. Undefined is not the same as zero.

Important formulas

sin⁡θ=OH,cos⁡θ=AH,tan⁡θ=OA\sin\theta=\frac OH,\quad\cos\theta=\frac AH,\quad\tan\theta=\frac OA
csc⁡θ=HO,sec⁡θ=HA,cot⁡θ=AO\csc\theta=\frac HO,\quad\sec\theta=\frac HA,\quad\cot\theta=\frac AO
sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1
1+tan⁡2θ=sec⁡2θ,1+cot⁡2θ=csc⁡2θ1+\tan^2\theta=\sec^2\theta,\quad1+\cot^2\theta=\csc^2\theta

Three worked examples

Example 1

In a right triangle, the side opposite an acute angle θ\theta is 88 cm and the hypotenuse is 1717 cm. Find sine, cosine and tangent of θ\theta.

  1. By Pythagoras, the adjacent side is 172−82=225=15\sqrt{17^2-8^2}=\sqrt{225}=15 cm.
  2. Use the definitions with the same reference angle: sin⁡θ=8/17\sin\theta=8/17 and cos⁡θ=15/17\cos\theta=15/17.
  3. Then tan⁡θ=8/15\tan\theta=8/15. As a check, 64/289+225/289=164/289+225/289=1.

Answer: sin⁡θ=8/17\sin\theta=8/17, cos⁡θ=15/17\cos\theta=15/17, tan⁡θ=8/15\tan\theta=8/15.

Example 2

Evaluate 2sin⁡30∘+cos⁡60∘−tan⁡45∘2\sin30^\circ+\cos60^\circ-\tan45^\circ exactly.

  1. Substitute sin⁡30∘=1/2\sin30^\circ=1/2, cos⁡60∘=1/2\cos60^\circ=1/2 and tan⁡45∘=1\tan45^\circ=1.
  2. The expression becomes 2(1/2)+1/2−12(1/2)+1/2-1.
  3. Perform multiplication before addition: 1+1/2−1=1/21+1/2-1=1/2.

Answer: 1/21/2.

Example 3

For an acute angle θ\theta, tan⁡θ=5/12\tan\theta=5/12. Find sec⁡θ\sec\theta and check the identity.

  1. Choose opposite and adjacent sides proportional to 55 and 1212. The hypotenuse is 25+144=13\sqrt{25+144}=13.
  2. Therefore sec⁡θ=H/A=13/12\sec\theta=H/A=13/12.
  3. Check 1+tan⁡2θ=1+25/144=169/144=(13/12)21+\tan^2\theta=1+25/144=169/144=(13/12)^2.

Answer: sec⁡θ=13/12\sec\theta=13/12.

Common errors to avoid

  • Do not identify opposite and adjacent before choosing the reference angle.
  • Write sin⁡2θ=(sin⁡θ)2\sin^2\theta=(\sin\theta)^2; it is not sin⁡(θ2)\sin(\theta^2).
  • Only cancel common factors, not terms in an addition.
  • Check denominator restrictions before dividing an identity by sine or cosine.

Try it yourself

Attempt these original questions before checking the answer and reasoning.

  1. An acute angle has sin⁡θ=7/25\sin\theta=7/25. Find cos⁡θ\cos\theta.

    Check answer and reasoning

    24/2524/25.

    The remaining side is 625−49=24\sqrt{625-49}=24; cosine is positive for an acute angle.

  2. Evaluate tan⁡30∘tan⁡60∘\tan30^\circ\tan60^\circ.

    Check answer and reasoning

    11.

    (1/3)(3)=1(1/\sqrt3)(\sqrt3)=1.

  3. Simplify (1−sin⁡2θ)/cos⁡2θ(1-\sin^2\theta)/\cos^2\theta where cos⁡θ≠0\cos\theta\ne0.

    Check answer and reasoning

    11.

    The numerator is cos⁡2θ\cos^2\theta, so numerator and denominator are equal and non-zero.

Related chapter guides

Quadratic Equations · Arithmetic Progressions

Use the algebra guides to strengthen rearrangement and substitution before simplifying trigonometric expressions.

Explore all Class 10 Maths chapters

Official syllabus and draft review

Aligned to the official CBSE Class X Mathematics syllabus for 2026–27 (041 and 241). Check the CBSE curriculum portal for updates and your school's guidance.

Original Mathslume study drafts, checked against the syllabus with independently calculated answers. Not official CBSE/NCERT solutions or an endorsement. A qualified teacher's editorial sign-off is still recommended before publication.

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