Trigonometry • Topic 1 of 3

Trigonometric Ratios and Identities

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

Solved Examples

Worked Example

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).

Solution

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.

Worked Example

If $\sin \theta$ = 5 / 13, calculate the exact fractional value of $\tan \theta$.

Solution

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.

Worked Example

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

Solution

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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Key Points

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 — 4 MCQs

Tap an option to check your answer0 / 4

Q1.$\sin\theta=$

Explanation: Sine $=$ opposite over hypotenuse.

Q2.$\sin^2\theta+\cos^2\theta=$

Explanation: Fundamental identity.

Q3.$\tan\theta=$

Explanation: $\tan=\sin/\cos$.

Q4.$1+\tan^2\theta=$

Explanation: Pythagorean identity.

Practice more MCQs → 📝 Assignment · 35 marks