Class 10 Maths · Original EZT practice
Introduction to Trigonometry (Class 10 Maths Chapter 8): six ratios, one table, one identity
Measure one angle and one side of a right triangle, and trigonometry hands you every other side - without a ruler. This chapter builds the three main ratios, the five standard values worth memorising, and the identity that ties sine and cosine together. Below: the ideas in plain words, then original EZT practice with worked answers. (Note: this is original practice, not a textbook exercise key.)
10 original worked examples, 10 fresh self-check questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 17 October 2026:
1. The three main ratios
In a right triangle, pick an acute angle A. Then: sin A = opposite / hypotenuse; cos A = adjacent / hypotenuse; tan A = opposite / adjacent. The ratios depend only on the angle, not on the triangle's size.
2. Standard values: the five-angle table
Worth memorising: sin 30 = 1/2, sin 45 = 1/sqrt2, sin 60 = sqrt3/2; cos runs the same values in reverse (cos 30 = sqrt3/2, cos 45 = 1/sqrt2, cos 60 = 1/2); tan 30 = 1/sqrt3, tan 45 = 1, tan 60 = sqrt3. At the extremes: sin 0 = 0, sin 90 = 1.
3. The identity sin2 A + cos2 A = 1
By Pythagoras, (opposite)^2 + (adjacent)^2 = (hypotenuse)^2; divide through by (hypotenuse)^2 and you get sin2 A + cos2 A = 1 - true for every angle. It is the fastest way to get cosine from sine (or the reverse).
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
Find sin, cos and tan from a triangle
Choose the method: Name opposite, adjacent and hypotenuse for the chosen angle.
EZT practice question
A right triangle has sides 3 and 4 with hypotenuse 5. For the angle opposite the side 3, find sin, cos and tan.
Show the worked solution
- Opposite = 3, adjacent = 4 and hypotenuse = 5.
- Use sin = opposite/hypotenuse, cos = adjacent/hypotenuse and tan = opposite/adjacent.
Answer: sin A = 3/5; cos A = 4/5; tan A = 3/4.
Check: The hypotenuse is the denominator for sin and cos, not for tan.
Watch for this: Do not swap adjacent and opposite after choosing the angle.
Now try it yourself
For the same triangle, find sin, cos and tan for the angle opposite the side 4.
Check your answer
sin = 4/5, cos = 3/5 and tan = 4/3.
The opposite and adjacent sides swap for the other acute angle.
Build the triangle from tan A
Choose the method: Turn tan A into an opposite-to-adjacent ratio, then use Pythagoras.
EZT practice question
If tan A = 4/3, find sin A and cos A.
Show the worked solution
- tan A = opposite/adjacent = 4/3.
- The 3-4-5 right triangle gives hypotenuse 5.
Answer: Opposite : adjacent = 4 : 3, so the hypotenuse is 5 (3-4-5 triangle): sin A = 4/5, cos A = 3/5.
Check: tan A = (4/5)/(3/5) = 4/3.
Watch for this: The side 4 is opposite, not the hypotenuse.
Now try it yourself
If tan A = 3/4, find sin A and cos A.
Check your answer
sin A = 3/5 and cos A = 4/5.
The opposite and adjacent are 3 and 4 in a 3-4-5 triangle.
Pick the adjacent side for cosine
Choose the method: Mark the side touching the chosen angle, excluding the hypotenuse.
EZT practice question
In a 6-8-10 right triangle, find the cosine of the angle between the hypotenuse and the side of 8.
Show the worked solution
- The side of 8 touches the chosen angle and is adjacent.
- The hypotenuse is 10, so cos = adjacent/hypotenuse.
Answer: Adjacent = 8, hypotenuse = 10, so cos = 8/10 = 4/5.
Check: The side of 6 is opposite this angle and is not used for cosine.
Watch for this: The hypotenuse is not an adjacent side.
Now try it yourself
In the same triangle, what is sin of that angle?
Check your answer
3/5.
The opposite side is 6, and 6/10 = 3/5.
Read an acute angle from the standard table
Choose the method: Match the given sine value to the memorised table.
EZT practice question
If sin A = 1/2 and A is acute, what is A?
Show the worked solution
- The standard table gives sin 30 = 1/2.
- A is acute, so the matching angle is 30 degrees.
Answer: A = 30 degrees.
Check: 30 degrees is acute and has sine 1/2.
Watch for this: Do not choose 150 degrees because A is stated to be acute.
Now try it yourself
If sin A = sqrt3/2 and A is acute, what is A?
Check your answer
60 degrees.
The standard table gives sin 60 = sqrt3/2.
Evaluate a standard-value sum
Choose the method: Substitute the table values before simplifying.
EZT practice question
Evaluate sin 60 x cos 30 + cos 60 x sin 30.
Show the worked solution
- sin 60 = sqrt3/2, cos 30 = sqrt3/2, cos 60 = 1/2 and sin 30 = 1/2.
- The products are 3/4 and 1/4.
Answer: (sqrt3/2)(sqrt3/2) + (1/2)(1/2) = 3/4 + 1/4 = 1.
Check: This is also sin(60 + 30) = sin 90 = 1.
Watch for this: Multiply each pair first; do not add the angles before substituting values.
Now try it yourself
Evaluate sin 30 x cos 60 + cos 30 x sin 60.
Check your answer
1.
The products are 1/4 and 3/4.
Evaluate a standard-value product
Choose the method: Replace each ratio with its exact table value.
EZT practice question
Evaluate tan 45 x sin 30 x cos 60.
Show the worked solution
- tan 45 = 1, sin 30 = 1/2 and cos 60 = 1/2.
- Multiply the three values.
Answer: 1 x 1/2 x 1/2 = 1/4.
Check: The two halves make one quarter, and multiplication by 1 changes nothing.
Watch for this: Do not add the fractions in a product question.
Now try it yourself
Evaluate tan 45 x cos 30 x sin 30.
Check your answer
sqrt3/4.
The product is 1 x sqrt3/2 x 1/2.
Check tan using sin over cos
Choose the method: Divide the sine value by the cosine value for the same angle.
EZT practice question
Check tan 30 against sin 30 / cos 30.
Show the worked solution
- sin 30 = 1/2 and cos 30 = sqrt3/2.
- Divide: (1/2)/(sqrt3/2) = 1/sqrt3.
Answer: (1/2) / (sqrt3/2) = 1/sqrt3 - the table value, since tan A = sin A / cos A always.
Check: The common denominator of 2 cancels.
Watch for this: Use the same angle for sin and cos.
Now try it yourself
Check tan 60 using sin 60 / cos 60.
Check your answer
sqrt3.
(sqrt3/2)/(1/2) = sqrt3.
Verify the main identity at 30 degrees
Choose the method: Square the standard sine and cosine values, then add.
EZT practice question
Verify the identity for A = 30 degrees.
Show the worked solution
- sin 30 = 1/2, so sin2 30 = 1/4.
- cos 30 = sqrt3/2, so cos2 30 = 3/4.
Answer: sin2 30 + cos2 30 = 1/4 + 3/4 = 1.
Check: The numerator parts add to 4 over a denominator of 4.
Watch for this: Square each value before adding; do not square the final sum.
Now try it yourself
Verify the identity for A = 45 degrees.
Check your answer
1/2 + 1/2 = 1.
sin 45 and cos 45 are both 1/sqrt2.
Find cosine from sine with the identity
Choose the method: Rearrange sin2 A + cos2 A = 1 and take the positive root for an acute angle.
EZT practice question
If sin A = 3/5, find cos A without drawing the triangle.
Show the worked solution
- cos2 A = 1 - sin2 A = 1 - 9/25 = 16/25.
- A is acute in the right-triangle setting, so cos A = 4/5.
Answer: cos2 A = 1 - 9/25 = 16/25, so cos A = 4/5.
Check: 3/5, 4/5 and 1 form the familiar 3-4-5 ratio triangle.
Watch for this: Do not forget the square root after finding cos2 A.
Now try it yourself
If sin A = 5/13, find cos A.
Check your answer
12/13.
cos2 A = 1 - 25/169 = 144/169.
Rearrange the identity
Choose the method: Move one squared term across the equals sign.
EZT practice question
Show that 1 - sin2 A equals cos2 A.
Show the worked solution
- Start with sin2 A + cos2 A = 1.
- Subtract sin2 A from both sides.
Answer: Rearrange sin2 A + cos2 A = 1 by moving sin2 A to the right - done; every identity question starts from this one line.
Check: Adding sin2 A back to cos2 A returns 1.
Watch for this: This is a rearrangement, not a new identity to memorise separately.
Now try it yourself
What does 1 - cos2 A equal?
Check your answer
sin2 A.
Subtract cos2 A from both sides of the main identity.
Try it yourself
Work these on paper before checking any notes. Answers are deliberately not shown here.
- In a 5-12-13 triangle, find sin and tan of the angle opposite the side 5.
- Evaluate 2 sin 30 + cos 60.
- If cos A = 12/13, find sin A.
- Is sin 60 = 2 sin 30? Check with the table.
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Questions about this practice page
- What are sin, cos and tan?
- Ratios of sides in a right triangle - sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent - fixed by the angle alone.
- What are the standard trigonometric values?
- sin 30 = 1/2, sin 45 = 1/sqrt2, sin 60 = sqrt3/2, with cos in reverse order, and tan 30 = 1/sqrt3, tan 45 = 1, tan 60 = sqrt3.
- What is the main trigonometric identity?
- sin2 A + cos2 A = 1 - it comes straight from Pythagoras and lets you find cosine from sine, or simplify expressions.
- Why does the triangle's size not matter?
- Enlarging a right triangle scales all its sides together, so every ratio - opposite over hypotenuse and the rest - stays exactly the same for the same angle.
- Are these the NCERT exercise questions for Introduction to Trigonometry?
- No. EZT pages are original practice on this chapter - use them to prepare for the textbook exercises, not as a substitute for your book.