IB Mathematics AA Reciprocal trigonometric ratios Study Notes | New Syllabus
IB Mathematics AA Reciprocal trigonometric ratios Study Notes
IB Mathematics AA Reciprocal trigonometric ratios Study Notes Offer a clear explanation of Reciprocal trigonometric ratios , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Reciprocal trigonometric ratios.
Reciprocal Trigonometric Ratios
Reciprocal Trigonometric Ratios
The reciprocal trigonometric ratios are defined as follows for angle \( \theta \) in a right-angled triangle:
Secant (sec θ): the reciprocal of cosine
\( \sec \theta = \frac{1}{\cos \theta} \) Defined for all \( \theta \) where \( \cos \theta \ne 0 \).
Cosecant (cosec θ or csc θ): the reciprocal of sine
\( \csc \theta = \frac{1}{\sin \theta} \) Defined for all \( \theta \) where \( \sin \theta \ne 0 \).
Cotangent (cot θ): the reciprocal of tangent
\( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \) Defined for all \( \theta \) where \( \tan \theta \ne 0 \), or equivalently where \( \sin \theta \ne 0 \).
Note: These functions are undefined at angles where their corresponding primary ratios are zero (e.g., \( \sec \theta \) is undefined at \( \theta = \frac{\pi}{2} \), \( \frac{3\pi}{2} \), etc.).
Example
Given that \( \sin \theta = \frac{3}{5} \) and \( \theta \) is in the first quadrant, find the values of \( \csc \theta \), \( \sec \theta \), and \( \cot \theta \).
▶️Answer/Explanation
Since \( \sin \theta = \frac{3}{5} \), we can draw a right-angled triangle where opposite = 3 and hypotenuse = 5.
Using Pythagoras’ theorem: adjacent = \( \sqrt{5^2 – 3^2} = \sqrt{25 – 9} = \sqrt{16} = 4 \).
\( \csc \theta = \frac{1}{\sin \theta} = \frac{5}{3} \)
\( \sec \theta = \frac{1}{\cos \theta} = \frac{1}{4/5} = \frac{5}{4} \)
\( \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{4/5}{3/5} = \frac{4}{3} \)
Pythagorean Identities
Pythagorean Identities
These identities arise from dividing the fundamental identity \( \sin^2 \theta + \cos^2 \theta = 1 \) by either \( \cos^2 \theta \) or \( \sin^2 \theta \).
Identity 1: Dividing by \( \cos^2 \theta \):
\( 1 + \tan^2 \theta = \sec^2 \theta \) Valid for all \( \theta \) where \( \cos \theta \ne 0 \).
Identity 2: Dividing by \( \sin^2 \theta \):
\( 1 + \cot^2 \theta = \csc^2 \theta \) Valid for all \( \theta \) where \( \sin \theta \ne 0 \).
Note: These identities are useful in simplifying expressions and solving trigonometric equations.
Example
If \( \tan \theta = 3 \) and \( \theta \) is in the first quadrant, find the value of \( \sec \theta \).
▶️Answer/Explanation
We know the identity: \( 1 + \tan^2 \theta = \sec^2 \theta \)
Substitute \( \tan \theta = 3 \): \( 1 + 3^2 = \sec^2 \theta \)
\( 1 + 9 = \sec^2 \theta \)
\( \sec^2 \theta = 10 \)
\( \sec \theta = \sqrt{10} \) (Since \( \theta \) is in the first quadrant, \( \sec \theta > 0 \))
Inverse Trigonometric Functions
Inverse Trigonometric Functions
Inverse Sine: \( f(x) = \arcsin x \)
Domain: \( -1 \le x \le 1 \)
Range: \( -\frac{\pi}{2} \le y \le \frac{\pi}{2} \)
Inverse Cosine: \( f(x) = \arccos x \)
Domain: \( -1 \le x \le 1 \)
Range: \( 0 \le y \le \pi \)
Inverse Tangent: \( f(x) = \arctan x \)
Domain: \( -\infty < x < \infty \)
Range: \( -\frac{\pi}{2} < y < \frac{\pi}{2} \)
Graphs: The graphs of these inverse functions are reflections of the graphs of \( \sin x \), \( \cos x \), and \( \tan x \) (restricted to their principal domains) in the line \( y = x \).
Important: These domain and range restrictions ensure that the inverse functions are one-to-one and thus valid functions.
Example
Find \( \theta \) if \( \arcsin \frac{1}{2} = \theta \). State \( \theta \) in radians.
▶️Answer/Explanation
\( \arcsin \frac{1}{2} = \theta \)
By definition, \( \theta = \arcsin \frac{1}{2} \) means \( \sin \theta = \frac{1}{2} \)
Since \( \arcsin x \) gives the angle in the range \( -\frac{\pi}{2} \le \theta \le \frac{\pi}{2} \):
\( \theta = \frac{\pi}{6} \)