Introduction to Trigonometry — Class 10 IMO

Trigonometric Ratios

What is Trigonometry? The word trigonometry comes from Greek words meaning "triangle measuring." It is a branch of mathematics that studies the relationship between the side lengths and angles of triangles. Imagine you are standing near a tall mobile tower and looking up at its top. If you know your distance from the base of the tower and the angle at which you look up, trigonometry helps you find the height of the tower without physically climbing up to measure it!

Trigonometric Ratios In a right-angled triangle, we name the sides relative to a specific acute angle, which we call theta (written as a special symbol). The sides are:

Hypotenuse: The longest side, directly opposite the 90-degree right angle.
Opposite side: The side directly facing our chosen angle theta.
Adjacent side: The side that runs alongside our angle theta and touches the right angle.

The six fundamental trigonometric ratios are simple fractions created by dividing the length of one side by another:

Ratio Name	Short Form	Fractional Formula	Reciprocal Partner
Sine	sin	Opposite / Hypotenuse	cosec = 1 / sin
Cosine	cos	Adjacent / Hypotenuse	sec = 1 / cos
Tangent	tan	Opposite / Adjacent	cot = 1 / tan
Cosecant	cosec	Hypotenuse / Opposite	sin = 1 / cosec
Secant	sec	Hypotenuse / Adjacent	cos = 1 / sec
Cotangent	cot	Adjacent / Opposite	tan = 1 / cot

Trigonometric Identities A trigonometric identity is an equation involving these ratios that stays perfectly true for every single angle value you can choose. The three core formulas are rooted in the Pythagoras theorem:

\(\sin^{2}\theta\) + \(\cos^{2}\theta\) = 1
1 + \(\tan^{2}\theta\) = \(\sec^{2}\theta\)
1 + \(\cot^{2}\theta\) = \(\cosec^{2}\theta\)

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DIAGRAM 1: NAMING SIDES RELATIVE TO ANGLE THETA

             |\
             | \
             |  \  Hypotenuse
    Opposite |   \  (Longest Side)
      Side   |    \
             |_____\ Angle Theta (θ)
             Right   Adjacent Side
             Angle

DIAGRAM 2: REAL-WORLD APPLICATION (HEIGHT OF A TOWER)

     Tower Top  X
                | \
                |  \ Line of Sight
    Tower Height|   \
                |____\ Observer Eyeball
               Base   Distance from Base

DIAGRAM 3: CORE RATIO AND IDENTITY FLOWCHART

     [ Pythagoras Theorem: Side1^2 + Side2^2 = Hypotenuse^2 ]
                                |
         +----------------------+----------------------+
         |                      |                      |
    (Divide by Hyp^2)      (Divide by Adj^2)      (Divide by Opp^2)
         |                      |                      |
         v                      v                      v
    sin^2 + cos^2 = 1      1 + tan^2 = sec^2      1 + cot^2 = cosec^2

Example 1: In a right-angled triangle ABC, the right angle is at vertex B. If the opposite side AB = 3 cm and the adjacent side BC = 4 cm, find the exact values of sin(A) and cos(A).

Step 1: Use the Pythagoras theorem to find the missing hypotenuse side AC.*
AC^2 = AB^2 + BC^2 = 3^2 + 4^2*
AC^2 = 9 + 16 = 25*
AC = Square root of 25 = 5 cm.*
Step 2: Determine sides relative to angle A.*
For angle A, the opposite side is BC = 4 cm, the adjacent side is AB = 3 cm, and the hypotenuse is AC = 5 cm.*
Step 3: Calculate sin(A) using the ratio formula.*
sin(A) = Opposite / Hypotenuse = 4 / 5.*
Step 4: Calculate cos(A) using the ratio formula.*
cos(A) = Adjacent / Hypotenuse = 3 / 5.*
Answer: sin(A) = 4/5 and cos(A) = 3/5.

Example 2: If \(\sin \theta\) = 5 / 13, calculate the exact fractional value of \(\tan \theta\).

Step 1: Relate the given fraction to the side definitions.*
\(\sin \theta\) = Opposite / Hypotenuse = 5 / 13.*
Let the Opposite side = 5 units and Hypotenuse = 13 units.*
Step 2: Calculate the missing Adjacent side using the Pythagoras theorem.*
Adjacent^2 = Hypotenuse^2 - Opposite^2 = 13^2 - 5^2*
Adjacent^2 = 169 - 25 = 144*
Adjacent = Square root of 144 = 12 units.*
Step 3: Compute \(\tan \theta\) using its unique formula.*
\(\tan \theta\) = Opposite / Adjacent = 5 / 12.*
Answer: \(\tan \theta\) = 5/12.

Example 3: Simplify the algebraic trigonometric expression: (1 + \(\tan^{2}\theta\)) * \(\cos^{2}\theta\).

Step 1: Identify an identity to substitute for the bracketed term.*
We know the standard identity: 1 + \(\tan^{2}\theta\) = \(\sec^{2}\theta\).*
Step 2: Substitute this identity directly into the starting expression.*
The expression becomes: \(\sec^{2}\theta\) \(\cos^{2}\theta\).
Step 3: Apply the reciprocal definition of the secant ratio.*
\(\sec \theta\) = 1 / \(\cos \theta\), which implies \(\sec^{2}\theta\) = 1 / \(\cos^{2}\theta\).*
Step 4: Perform the final algebraic simplification.*
(1 / \(\cos^{2}\theta\)) \(\cos^{2}\theta\) = 1.
Answer: 1.
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Quick recap

Trigonometry links the values of interior acute angles to the ratio of side lengths in right triangles.
The three primary ratios are sin (Opp/Hyp), cos (Adj/Hyp), and tan (Opp/Adj).
cosec, sec, and cot are the direct multiplicative reciprocals of sin, cos, and tan.
The square identity \(\sin^{2}\theta\) + \(\cos^{2}\theta\) = 1 is derived directly from the Pythagoras theorem.
Tangent can also be expressed as a quotient: \(\tan \theta\) = \(\sin \theta\) / \(\cos \theta\).

✓ Quick check

If cos θ = 1/2, then θ equals:

cos 60° = 1/2, so θ = 60°.

The value of cos 0° is:

cos 0° = 1.

Identities and Complementary Angles

What are Complementary Angles in Trigonometry? Two angles are called complementary angles if their sum equals exactly 90 degrees. In any right-angled triangle, since the right angle uses up 90 degrees, the remaining two acute angles must add up to 90 degrees. They are always complementary pairs!

Because these two angles share the same right triangle, the opposite side for one angle automatically becomes the adjacent side for the other angle. This introduces a set of formulas called co-ratio properties:

sin(90° - theta) = \(\cos \theta\)
cos(90° - theta) = \(\sin \theta\)
tan(90° - theta) = \(\cot \theta\)
cot(90° - theta) = \(\tan \theta\)
sec(90° - theta) = \(\cosec \theta\)
cosec(90° - theta) = \(\sec \theta\)

Think of it as a tag team match: Sine pairs with Cosine, Tangent pairs with Cotangent, and Secant pairs with Cosecant. When an angle is subtracted from 90°, the ratio switches to its corresponding partner!

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DIAGRAM 1: COMPLEMENTARY ANGLES IN A RIGHT TRIANGLE

             |\ Angle = (90° - θ)
             | \
             |  \
             |   \
             |_____\ Angle = θ
             Right
             Angle

DIAGRAM 2: THE CO-RATIO PAIRING SYSTEM

       [ Sine ]       <====== Complementary ======>       [ Cosine ]
       [ Tangent ]    <====== Complementary ======>       [ Cotangent ]
       [ Secant ]     <====== Complementary ======>       [ Cosecant ]

DIAGRAM 3: CONVERSION EXAMPLE PROCESS

      Given Term:  cos(53°)
      Step 1: Rewrite angle as a difference:  cos(90° - 37°)
      Step 2: Apply complementary identity:   sin(37°)

Example 7: Evaluate the exact value of the fractional expression: sin(18°) / cos(72°).

Step 1: Check if the two given angles are complementary.*
18° + 72° = 90°. Yes, they are complementary angles.*
Step 2: Convert one of the terms using a complementary angle formula.*
Let us convert the numerator: sin(18°) = sin(90° - 72°).*
Using the identity sin(90° - theta) = \(\cos \theta\), we find sin(90° - 72°) = cos(72°).*
Step 3: Substitute this value back into the original fraction.*
Expression = cos(72°) / cos(72°).*
Step 4: Simplify the identical division.*
cos(72°) / cos(72°) = 1.*
Answer: 1.

Example 8: Evaluate the following value without a table: tan(48°) tan(23°) tan(42°) * tan(67°).

Step 1: Group the angles into complementary pairs.*
(48° + 42° = 90°) and (23° + 67° = 90°).*
Rearrange the expression: [tan(48°) tan(42°)] [tan(23°) tan(67°)].
Step 2: Convert one tangent in each pair to a cotangent.*
tan(42°) = tan(90° - 48°) = cot(48°).*
tan(67°) = tan(90° - 23°) = cot(23°).*
Step 3: Rewrite the multiplication using these conversions.*
[tan(48°) cot(48°)] [tan(23°) cot(23°)].
Step 4: Apply the reciprocal rule: \(\tan \theta\) \(\cot \theta\) = 1.
[1] [1] = 1.
Answer: 1.

Example 9: If sin(3A) = cos(A - 26°), where 3A is an acute angle, find the exact degree value of A.

Step 1: Use the complementary formula to make the ratios on both sides identical.*
We know that sin(3A) can be written as cos(90° - 3A).*
Step 2: Substitute this equivalent term into the given equation.*
cos(90° - 3A) = cos(A - 26°).*
Step 3: Equate the two angles since their cosine values match.*
90° - 3A = A - 26°*
Step 4: Group like terms and solve for angle A.*
90° + 26° = A + 3A*
116° = 4A*
A = 116° / 4 = 29°.*
Answer: 29°.
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Quick recap

Complementary acute angles always add up to a combined sum of 90 degrees.
The function \(\sin \theta\) converts directly into cos(90° - theta).
The function \(\tan \theta\) pairs complements with cot(90° - theta).
The function \(\sec \theta\) changes over into cosec(90° - theta).
When evaluating complex products, look to pair values whose angles add up to 90 degrees.

✓ Quick check

The identity 1 + cot²θ equals:

1 + cot²θ = cosec²θ.

If tan θ = 1, then θ equals:

tan 45° = 1, so θ = 45°.

Heights and Distances

An angle of elevation looks up to an object; an angle of depression looks down. Model the situation as a right triangle and use tan, sin or cos to find the unknown height or distance.

Example 1: Elevation 45° to a 100 m tower — distance?

tan 45° = 100/d = 1, so d = 100 m.

Example 2: A tower's shadow equals its height — sun's elevation?

tan θ = 1, so 45°.

Quick recap

Elevation looks up, depression looks down (equal alternate angles).
Use a right-triangle ratio to find the unknown.

✓ Quick check

An aeroplane is flying at a height of 1000 m. Its angle of elevation from a ground point is 30°. The horizontal distance of the point from the plane is:

tan 30° = 1000/d, so d = 1000√3 m.

A tower casts a shadow exactly equal in length to its height. The sun's angle of elevation is:

tan θ = height/shadow = 1, so θ = 45°.