Leaf decoration

Trigonometry

Unit: 3
Book Icon

Class 9: Optional Math

Trigonometric Ratios, Trigonometric Ratios of Special Angles, Identities of Trigonometric Ratios, Trigonometric Ratios of Any Angles, Trigonometric Ra...

AI-Powered
TL;DR — Quick Summary
Click Generate Summary to get a quick AI-powered overview of this chapter.
Gemini is reading the chapter...
    Could not generate summary. Please try again.
    Explain This
    AI Explanation
    Explaining...

    Could not explain. Try again.
    MCQ Practice
    Generate MCQ

    Trigonometric Ratios

    These ratios are the fundamental relationships in trigonometry and are used to relate the angles of a triangle to the lengths of its sides.

    Basic Trigonometric Ratios: sin, cos, tan

    Reciprocal Trigonometric Ratios: cosec, sec, cot

    For a given angle \(\theta\):

    - Perpendicular: The side opposite the angle \(\theta\). (Opposite)

    - Base: The side next to the angle \(\theta\). (Adjacent Side)

    - Hypotenuse: The longest side, opposite the right angle.

     

    There are six trigonometric ratios:

    1. Sine (sin): The ratio of the length of the Perpendicular to the Hypotenuse.

    2. Cosine (cos): The ratio of the length of the Base to the Hypotenuse.

    3. Tangent (tan): The ratio of the length of the Perpendicular to the Base.

    4. Cosecant (cosec/csc): The reciprocal of sine. It is the ratio of the Hypotenuse to the Perpendicular.

    5. Secant (sec): The reciprocal of cosine. It is the ratio of the Hypotenuse to the Base.

    6. Cotangent (cot): The reciprocal of tangent. It is the ratio of the Base to the Perpendicular.

    Trigonometric Ratios

    Sine
    $$\sin\theta = \frac{p}{h}$$
    Cosine
    $$\cos\theta = \frac{b}{h}$$
    Tangent
    $$\tan\theta = \frac{p}{b}$$
    Cosecant
    $$\csc\theta = \frac{1}{\sin\theta} = \frac{h}{p}$$
    Secant
    $$\sec\theta = \frac{1}{\cos\theta} = \frac{h}{b}$$
    Cotangent
    $$\cot\theta = \frac{1}{\tan\theta} = \frac{b}{p}$$

     

    Trigonometric Ratio of Special Angles

    The values of trigonometric ratios for \(0^0\), \(30^0\), \(45^0\), \(60^0\), and \(90^0\), are as follows.

    Angle\[0^0\]\[30^0\]\[45^0\]\[60^0\]\[90^0\]
    Ratio
    \[\text{sin}\]

    0

    		\[\frac{1}{2}\]\[\frac{1}{\sqrt{2}}\]\[\frac{\sqrt{3}}{2}\]\[1\]
    \[\text{cos}\]\[1\]\[\frac{\sqrt{3}}{2}\]\[\frac{1}{\sqrt{2}}\]\[\frac{1}{2}\]\[0\]
    \[\text{tan}\]\[0\]\[\frac{1}{\sqrt{3}}\]\[1\]\[\sqrt{3}\]\[\infty\]

     

    Identities of Trigonometric Ratios

    What are identities?

    - In mathematics, Identities are equations that are true for all values of the variable(s) within their domain. Trigonometric identities are equalities involving trigonometric functions that hold true for all valid angles. These are essential for solving trigonometric equations, simplifying expressions, and proving relationships in geometry and calculus.

    Here are the Trigonometric Identities categorized by type:

    Trigonometric Identities

    Pythagorean
    \(\begin{aligned} & a. sin^{2}\theta + cos^{2}\theta = 1 \\ & b. 1 + tan^{2}\theta = sec^{2}\theta \\ & c. 1 + cot^{2}\theta = cosec^{2}\theta \end{aligned}\)
    Reciprocal
    \(\begin{aligned} & a. cosec \theta = \frac{1}{sin\theta} \\ & b. sec \theta = \frac{1}{cos\theta} \\ & c. cot \theta = \frac{1}{tan\theta} \end{aligned}\)
    Quotient
    \(\begin{aligned} & a. tan\theta = \frac{sin\theta}{cos\theta} \\ & b. cot\theta = \frac{cos\theta}{sin\theta} \\ \end{aligned}\)

     

    Typical Patterns of Identities in Trigonometry

    In this topic, we will learn about the standard patterns of trigonometric identities. Some examples include trigonometric identities containing the square root, implementation of algebraic formulas in trigonometry, etc.  Here are some worked out examples.

    \(\begin{aligned} & \text{Prove that. } \\ & \text{Q. 1 } \frac{tanA}{1 - cotA} + \frac{cotA}{1-tanA} = 1 + secAcscA \\ & \text{LHS } = \frac{tanA}{1 - cotA} + \frac{cotA}{1-tanA} \\ & = \frac{tanA}{1 - \frac{1}{tanA}} + \frac{\frac{1}{tanA}}{1-tanA} \\ & = \frac{tan^2A}{tanA - 1} + \frac{1}{tanA(1-tanA)} \\ & = \frac{tan^2A}{tanA - 1} - \frac{1}{tanA(tanA - 1)} \\ & = \frac{tan^3A - 1}{tanA(tanA - 1)} \\ & = \frac{(tanA – 1)(tan^2A + tanA + 1)}{tanA(tanA - 1)} \\ & = tanA + cotA + 1 \\ & = 1 + \frac{sinA}{cosA} + \frac{cosA}{sinA} \\ & = 1 + \frac{sin^2A + cos^2A}{sinAcosA} \\ & = 1 + \frac{1}{sinAcosA} \\ & \therefore \boxed{1 + secAcscA} = \text{ RHS } \text{Hence, Proved.} \end{aligned}\)

     

    \(\begin{aligned} & \text{Prove that.} \\ & \text{Q. 1} sin^8A – cos^8A = (2sin^2A - 1)(1-2sin^2A.cos^2A) \\ & \text{LHS} = sin^8A – cos^8A \\ & = (sin^4A)^2 – (cos^4A)^2 \\ & = (sin^4A – cos^4A)(sin^4A + cos^4A) \\ & = (sin^2A – cos^2A) (sin^2A + cos^2A)(sin^4A + cos^4A) \\ & = (sin^2A - 1 + sin^2A)(sin^4A + cos^4A) \\ & = (2sin^2A -1){(sin^2A)^2 + cos^2A)^2} \\ & = (2sin^2A - 1){(sin^2A + cos^2A)^2 – 2sin^2Acos^2A} \\ & \therefore \boxed{(2sin^2A - 1)(1 – 2sin^2Acos^2A)} = \text{ RHS, Hence, Proved.} \end{aligned}\)

     

    Trigonometric ratios of any angle

    In the previous chapter, we learned the Trigonometric Ratio of Special Angles like \(0^\circ , 30^\circ , 45^\circ , 60^\circ \text{and, } 90^\circ \) . Now in this chapter, we will explore techniques and methods for Trigonometric Ratios of Any Angles. Below is the list of trigonometric functions \( \sin, \cos, \tan, \csc, \sec, \cot \) for the given angles in terms of \( \theta \):

    \(\begin{aligned} & A. (90^\circ - \theta) \\ & 1. \sin(90^\circ - \theta) = \cos\theta \\ & 2. \cos(90^\circ - \theta) = \sin\theta \\ & 3. \tan(90^\circ - \theta) = \cot\theta \\ & 4. \csc(90^\circ - \theta) = \sec\theta \\ & 5. \sec(90^\circ - \theta) = \csc\theta \\ & 6. \cot(90^\circ - \theta) = \tan\theta \\ \\ \end{aligned}\)
    \(\begin{aligned} & B. (90^\circ + \theta) \\ & 1. \sin(90^\circ + \theta) = \cos\theta \\ & 2. \cos(90^\circ + \theta) = -\sin\theta \\ & 3. \tan(90^\circ + \theta) = -\cot\theta \\ & 4. \csc(90^\circ + \theta) = \sec\theta \\ & 5. \sec(90^\circ + \theta) = -\csc\theta \\ & 6. \cot(90^\circ + \theta) = -\tan\theta \\ \end{aligned}\)
    \(\begin{aligned} & C. ( 180^\circ - \theta ) \\ & 1. \sin(180^\circ - \theta) = \sin\theta \\ & 2. \cos(180^\circ - \theta) = -\cos\theta \\ & 3. \tan(180^\circ - \theta) = -\tan\theta \\ & 4. \csc(180^\circ - \theta) = \csc\theta \\ & 5. \sec(180^\circ - \theta) = -\sec\theta \\ & 6. \cot(180^\circ - \theta) = -\cot\theta \\ \\ \end{aligned} \)
    \(\begin{aligned} & D. (180^\circ + \theta ) \\ & 1. \sin(180^\circ + \theta) = -\sin\theta \\ & 2. \cos(180^\circ + \theta) = -\cos\theta \\ & 3. \tan(180^\circ + \theta) = \tan\theta \\ & 4. \csc(180^\circ + \theta) = -\csc\theta \\ & 5. \sec(180^\circ + \theta) = -\sec\theta \\ & 6. \cot(180^\circ + \theta) = \cot\theta \\ \\ \end{aligned} \)
    \(\begin{aligned} & E. ( 270^\circ - \theta ) \\ & 1. \sin(270^\circ - \theta) = -\cos\theta \\ & 2. \cos(270^\circ - \theta) = -\sin\theta \\ & 3. \tan(270^\circ - \theta) = \cot\theta \\ & 4. \csc(270^\circ - \theta) = -\sec\theta \\ & 5. \sec(270^\circ - \theta) = -\csc\theta \\ & 6. \cot(270^\circ - \theta) = \tan\theta \\ \\ \end{aligned} \)
    \(\begin{aligned} & F. ( 270^\circ + \theta ) \\ & 1. \sin(270^\circ + \theta) = -\cos\theta \\ & 2. \cos(270^\circ + \theta) = \sin\theta \\ & 3. \tan(270^\circ + \theta) = -\cot\theta \\ & 4. \csc(270^\circ + \theta) = -\sec\theta \\ & 5. \sec(270^\circ + \theta) = \csc\theta \\ & 6. \cot(270^\circ + \theta) = -\tan\theta \\ \\ \end{aligned} \)
    \(\begin{aligned} & G. ( 360^\circ - \theta) \\ & 1. \sin(360^\circ - \theta) = -\sin\theta \\ & 2. \cos(360^\circ - \theta) = \cos\theta \\ & 3. \tan(360^\circ - \theta) = -\tan\theta \\ & 4. \csc(360^\circ - \theta) = -\csc\theta \\ & 5. \sec(360^\circ - \theta) = \sec\theta \\ & 6. \cot(360^\circ - \theta) = -\cot\theta \\ \\ \end{aligned} \)
    \(\begin{aligned} & H. ( 360^\circ + \theta ) \\ & 1. \sin(360^\circ + \theta) = \sin\theta \\ & 2. \cos(360^\circ + \theta) = \cos\theta \\ & 3. \tan(360^\circ + \theta) = \tan\theta \\ & 4. \csc(360^\circ + \theta) = \csc\theta \\ & 5. \sec(360^\circ + \theta) = \sec\theta \\ & 6. \cot(360^\circ + \theta) = \cot\theta \\ \\ \end{aligned}\)

     

    Trigonometric Ratios of Compound Angles

    What is a Compound Angle? 

    A compound angle is the sum or difference of two angles. If \( A \) and \( B \) are two angles, then their sum \( (A + B) \) and difference \( (A - B) \) are called compound angles. 

    Formulae for Trigonometric Ratios of Compound Angles

    \(\begin{aligned} & \text{Formulae for Compound Angles:} \\ & 1. \sin(A + B) = \sin A \cos B + \cos A \sin B \\ & 2. \sin(A - B) = \sin A \cos B - \cos A \sin B \\ & 3. \cos(A + B) = \cos A \cos B - \sin A \sin B \\ & 4. \cos(A - B) = \cos A \cos B + \sin A \sin B \\ & 5. \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}, \quad \text{(if \( 1 - \tan A \tan B \neq 0 \))} \\ & 6. \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}, \quad \text{(if \( 1 + \tan A \tan B \neq 0 \))} \end{aligned}\)
    Generate MCQ

    Share Now

    Share to help more learners!

    Resources
    Lesson Contents