Sine and Cosine Are One Ratio Seen from the Other Corner
Use the complementary-angle relations to swap sine for cosine and tangent for cotangent, collapse expressions like sin 35 over cos 55 to a single number, and solve equations where a sine equals a cosine.
Why does the sine of one angle equal the cosine of another?
In a right-angled triangle the three angles add to and one of them is , so the two acute angles add to exactly :
Two angles that add to are called complementary, and every right-angled triangle contains such a pair.
Now look at one side — say the side . From 's corner it is the opposite side; from 's corner the very same side is the adjacent one. Nothing about the triangle changed; only the corner you are standing in did.
So
are the same fraction, which means
**Check it on the -- triangle.** There , and the other acute angle has as well. The two angles are about and , and they add to .
That is the whole content of this part of the chapter, and it has a practical payoff: half the trigonometric table is redundant. A table running from to is enough, because every angle above is the complement of one below it.
This page covers the third part of the ICSE Class 9 Mathematics chapter on trigonometrical ratios: the complementary-angle relations, using them to simplify, evaluating expressions without tables, and solving for an unknown angle.
Two angles that add to are called complementary, and every right-angled triangle contains such a pair.
Now look at one side — say the side . From 's corner it is the opposite side; from 's corner the very same side is the adjacent one. Nothing about the triangle changed; only the corner you are standing in did.
So
are the same fraction, which means
**Check it on the -- triangle.** There , and the other acute angle has as well. The two angles are about and , and they add to .
That is the whole content of this part of the chapter, and it has a practical payoff: half the trigonometric table is redundant. A table running from to is enough, because every angle above is the complement of one below it.
This page covers the third part of the ICSE Class 9 Mathematics chapter on trigonometrical ratios: the complementary-angle relations, using them to simplify, evaluating expressions without tables, and solving for an unknown angle.
Formula
What are the complementary-angle relations for all six ratios?
**Each ratio turns into its co partner when the angle is replaced by its complement.**
**The three pairs are exactly the three co pairs**: sine with cosine, tangent with cotangent, secant with cosecant. That is what the co in those names means — complement.
Where the tangent version comes from. In the same triangle, while , since the two sides swap roles from 's corner. So .
The secant and cosecant versions follow by taking reciprocals of the sine and cosine relations — nothing new is needed.
Worked conversions.
- , since
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Check one against the standard values. Taking :
and , as required. Likewise and .
One angle is its own complement. Since , the relation gives — which is exactly what the half-square triangle showed, with both legs equal. **So is the angle where sine and cosine cross**, and it is the reason .
The condition that must hold. These relations need the two angles to add to exactly . is not , because . Add the two angles before applying any of these six lines — it takes a second and it is the only check you need.
**The three pairs are exactly the three co pairs**: sine with cosine, tangent with cotangent, secant with cosecant. That is what the co in those names means — complement.
Where the tangent version comes from. In the same triangle, while , since the two sides swap roles from 's corner. So .
The secant and cosecant versions follow by taking reciprocals of the sine and cosine relations — nothing new is needed.
Worked conversions.
- , since
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Check one against the standard values. Taking :
and , as required. Likewise and .
One angle is its own complement. Since , the relation gives — which is exactly what the half-square triangle showed, with both legs equal. **So is the angle where sine and cosine cross**, and it is the reason .
The condition that must hold. These relations need the two angles to add to exactly . is not , because . Add the two angles before applying any of these six lines — it takes a second and it is the only check you need.
How do you simplify an expression using these relations?
Convert every ratio to the same function, then cancel. The usual move is to turn the larger angle into the smaller one, or every cosine into a sine.
Worked example 1. Simplify .
Since , we have , so
Worked example 2. Evaluate .
, and , so
Worked example 3. Evaluate .
Since , , and a tangent times its own cotangent is :
Worked example 4 — a longer product. Evaluate .
Pair the angles that add to : with , and with :
Look for the pairs before doing anything else. A product of four or six tangents is almost always built from complementary pairs, and the answer is almost always .
Worked example 5 — using the Pythagorean identity as well. Evaluate .
, so the expression becomes
Two chapters combined in one line — the complementary relation brought both terms to the same angle, and the identity from Part 1 finished it.
Worked example 6. Evaluate .
Take the terms one at a time. because . And because . So the value is
Worked example 7 — one that does not collapse. Simplify .
Here , not , so nothing cancels and no exact value is available without tables. Recognising that a question does not simplify is as important as simplifying one that does — and it is why the first step is always to add the two angles.
Worked example 1. Simplify .
Since , we have , so
Worked example 2. Evaluate .
, and , so
Worked example 3. Evaluate .
Since , , and a tangent times its own cotangent is :
Worked example 4 — a longer product. Evaluate .
Pair the angles that add to : with , and with :
Look for the pairs before doing anything else. A product of four or six tangents is almost always built from complementary pairs, and the answer is almost always .
Worked example 5 — using the Pythagorean identity as well. Evaluate .
, so the expression becomes
Two chapters combined in one line — the complementary relation brought both terms to the same angle, and the identity from Part 1 finished it.
Worked example 6. Evaluate .
Take the terms one at a time. because . And because . So the value is
Worked example 7 — one that does not collapse. Simplify .
Here , not , so nothing cancels and no exact value is available without tables. Recognising that a question does not simplify is as important as simplifying one that does — and it is why the first step is always to add the two angles.
How do you find the angle in an equation like sin x equals cos y?
**Set the two angles to add to and solve the resulting linear equation.** If , then and are complementary, so .
Worked example 1. Solve .
The two angles must be complementary:
Check: and , and because , as required.
Worked example 2 — tangent and cotangent. Solve .
Tangent pairs with cotangent, so again the angles are complementary:
Check: , , and , as required.
Worked example 3 — secant and cosecant. Solve .
Check: and , which add to , as required.
Worked example 4 — same function on both sides. Solve , with acute.
Here the functions match, so the angles themselves must be equal — the complementary rule does not apply:
That distinction is the whole skill. Same function on both sides means equal angles; a function and its co partner means complementary angles. Applying the wrong one is the commonest error in this section, and the cure is to name the two functions before writing any equation.
Worked example 5 — a cotangent written as a tangent. Solve .
Cotangent and tangent are co partners, so
Check: and , as required — and .
One boundary case to keep in mind. An answer must leave both angles between and , since these relations were proved inside a right-angled triangle. If solving gave in an equation containing , the angle would be negative and the answer would have to be rejected. Substitute back and check both angles are acute before writing the final line.
Worked example 1. Solve .
The two angles must be complementary:
Check: and , and because , as required.
Worked example 2 — tangent and cotangent. Solve .
Tangent pairs with cotangent, so again the angles are complementary:
Check: , , and , as required.
Worked example 3 — secant and cosecant. Solve .
Check: and , which add to , as required.
Worked example 4 — same function on both sides. Solve , with acute.
Here the functions match, so the angles themselves must be equal — the complementary rule does not apply:
That distinction is the whole skill. Same function on both sides means equal angles; a function and its co partner means complementary angles. Applying the wrong one is the commonest error in this section, and the cure is to name the two functions before writing any equation.
Worked example 5 — a cotangent written as a tangent. Solve .
Cotangent and tangent are co partners, so
Check: and , as required — and .
One boundary case to keep in mind. An answer must leave both angles between and , since these relations were proved inside a right-angled triangle. If solving gave in an equation containing , the angle would be negative and the answer would have to be rejected. Substitute back and check both angles are acute before writing the final line.
Exam tip
What layout keeps complementary-angle questions safe?
Add the two angles as your first written step. If they make , the relation applies; if they do not, nothing will cancel and the question is asking for something else.
- Write the conversion explicitly: * since .* The justification is where the mark sits
- Convert to one function before cancelling, usually turning every cosine into a sine or every larger angle into its complement
- Look for complementary pairs in a long product and bracket them:
- Remember that sine and cosine of the same angle also combine through — many questions need both chapters
- Name the two functions before solving an equation: matching functions give equal angles, co partners give complementary angles
- Substitute your answer back and confirm the two angles really add to
- Check both angles are acute and reject an answer that makes either negative
- Do not reach for tables — an ICSE question of this type always collapses to , or a standard value
The misconception to name. is not . Test it at : the left side is , while the right side would be . The bracket cannot be broken up, which is the same rule that stopped becoming in Part 2.
- Write the conversion explicitly: * since .* The justification is where the mark sits
- Convert to one function before cancelling, usually turning every cosine into a sine or every larger angle into its complement
- Look for complementary pairs in a long product and bracket them:
- Remember that sine and cosine of the same angle also combine through — many questions need both chapters
- Name the two functions before solving an equation: matching functions give equal angles, co partners give complementary angles
- Substitute your answer back and confirm the two angles really add to
- Check both angles are acute and reject an answer that makes either negative
- Do not reach for tables — an ICSE question of this type always collapses to , or a standard value
The misconception to name. is not . Test it at : the left side is , while the right side would be . The bracket cannot be broken up, which is the same rule that stopped becoming in Part 2.
Did you know
Why did a table from zero to forty-five degrees serve for everything?
Turn to the trigonometric tables at the back of a textbook and you will find columns of sines, cosines and tangents. What is less obvious is that half of that printing is unnecessary.
Because , a single column of numbers can be read two ways: down the page as sines of angles increasing from , and up the page as cosines of angles decreasing from . That is exactly how a four-figure table is laid out, with the sine angles printed down the left margin and the cosine angles up the right.
**So a table need only run from to .** Every angle above is the complement of one below it, and the entry is already there:
The same halving shows up in the slide rule and in the scales on a set square, and it is the reason trigonometric tables were ever small enough to print in the back of a school book at all.
**There is a neater way to see why is the turning point.** As an angle grows from to its sine rises from to while its cosine falls from to . The two must cross somewhere, and they cross where the angle equals its own complement:
At that crossing the triangle is isosceles, both legs are equal, and . Everything in this part of the chapter is a consequence of that one symmetry — the diagonal of a square being the place where opposite and adjacent stop being different.
Because , a single column of numbers can be read two ways: down the page as sines of angles increasing from , and up the page as cosines of angles decreasing from . That is exactly how a four-figure table is laid out, with the sine angles printed down the left margin and the cosine angles up the right.
**So a table need only run from to .** Every angle above is the complement of one below it, and the entry is already there:
The same halving shows up in the slide rule and in the scales on a set square, and it is the reason trigonometric tables were ever small enough to print in the back of a school book at all.
**There is a neater way to see why is the turning point.** As an angle grows from to its sine rises from to while its cosine falls from to . The two must cross somewhere, and they cross where the angle equals its own complement:
At that crossing the triangle is isosceles, both legs are equal, and . Everything in this part of the chapter is a consequence of that one symmetry — the diagonal of a square being the place where opposite and adjacent stop being different.
Exam relevance
How are complementary-angle relations used in JEE questions?
This is foundation work that becomes a tool for collapsing expressions rather than a topic in itself.
Where it leads. In Class 11 Trigonometric Functions these six relations are absorbed into the general allied-angle results, which extend them to , , and so on, with signs depending on the quadrant. The complementary case you learn here is the one that keeps the co swap, and it is the pattern the rest are built on. JEE Main sets simplification questions where the first step is precisely to pair complementary angles.
Where the collapsing trick reappears. Long products and sums that reduce to or are a standard JEE Main item, usually dressed as a sum like over angles that pair up. In Inverse Trigonometric Functions the relation appears as
which is the same statement about complements, written for inverse functions — and that identity is examined directly.
Where it appears in Physics. When a vector is resolved along two perpendicular directions, the two components are and — and they are interchangeable depending on which axis you call the reference. **The complementary relation is why along one axis is along the other, which is the source of many sign-and-function mix-ups in inclined-plane questions for JEE and NEET.
Question types to expect. At this level: simplify, evaluate without tables, solve for an angle. In competitive papers: collapse a product or sum, and inverse-trigonometric identities. Assertion-reason items test the condition** — whether the two angles actually add to .
The single trap that costs marks. Applying the relation to angles that are not complementary. is not , and in Class 11 the equivalent error is using the complementary rule where an allied-angle rule with a sign change is needed. Add the angles first, every time.
Board versus competitive emphasis. ICSE wants the conversion line and its justification; a competitive paper wants the collapsed value. **The transferable habit is scanning for pairs that add to ** before attempting any algebra — in a long expression that scan is usually the entire solution.
Where it leads. In Class 11 Trigonometric Functions these six relations are absorbed into the general allied-angle results, which extend them to , , and so on, with signs depending on the quadrant. The complementary case you learn here is the one that keeps the co swap, and it is the pattern the rest are built on. JEE Main sets simplification questions where the first step is precisely to pair complementary angles.
Where the collapsing trick reappears. Long products and sums that reduce to or are a standard JEE Main item, usually dressed as a sum like over angles that pair up. In Inverse Trigonometric Functions the relation appears as
which is the same statement about complements, written for inverse functions — and that identity is examined directly.
Where it appears in Physics. When a vector is resolved along two perpendicular directions, the two components are and — and they are interchangeable depending on which axis you call the reference. **The complementary relation is why along one axis is along the other, which is the source of many sign-and-function mix-ups in inclined-plane questions for JEE and NEET.
Question types to expect. At this level: simplify, evaluate without tables, solve for an angle. In competitive papers: collapse a product or sum, and inverse-trigonometric identities. Assertion-reason items test the condition** — whether the two angles actually add to .
The single trap that costs marks. Applying the relation to angles that are not complementary. is not , and in Class 11 the equivalent error is using the complementary rule where an allied-angle rule with a sign change is needed. Add the angles first, every time.
Board versus competitive emphasis. ICSE wants the conversion line and its justification; a competitive paper wants the collapsed value. **The transferable habit is scanning for pairs that add to ** before attempting any algebra — in a long expression that scan is usually the entire solution.
Key takeaways
What should you be able to do with complementary angles?
One fact about right-angled triangles, applied six ways.
- The two acute angles of a right-angled triangle are complementary, adding to
- What is opposite one angle is adjacent to the other, which is why
- **The three co pairs**: sine with cosine, tangent with cotangent, secant with cosecant — each swaps under
- Add the two angles first. The relation applies only when they make exactly
- To simplify, convert to one function and cancel; a ratio over its own complement is
- Pair complementary angles in a long product — the answer is usually or
- **Combine with when two terms can be brought to the same angle
- In an equation, matching functions give equal angles; co partners give complementary angles
- is its own complement**, which is why and
- ** is not ** — the bracket stays whole
The quickest self-test is the four-tangent product: evaluate in one line, then try — and notice which chapter each one needed.
- The two acute angles of a right-angled triangle are complementary, adding to
- What is opposite one angle is adjacent to the other, which is why
- **The three co pairs**: sine with cosine, tangent with cotangent, secant with cosecant — each swaps under
- Add the two angles first. The relation applies only when they make exactly
- To simplify, convert to one function and cancel; a ratio over its own complement is
- Pair complementary angles in a long product — the answer is usually or
- **Combine with when two terms can be brought to the same angle
- In an equation, matching functions give equal angles; co partners give complementary angles
- is its own complement**, which is why and
- ** is not ** — the bracket stays whole
The quickest self-test is the four-tangent product: evaluate in one line, then try — and notice which chapter each one needed.