6.2 Trigonometric Ratios of any Angle
This topic introduces the reciprocal trigonometric ratios—cosecant, secant, and cotangent—and extends the evaluation of trigonometric ratios to any angle in the Cartesian plane. Students learn how to use complementary angle identities, determine reference angles across all four quadrants, assign correct signs using coordinate systems or quadrants, and find exact values using special angles and unit circles.
Learning Objectives
- Relate secant, cosecant, and cotangent with sine, cosine, and tangent of any angle in a Cartesian plane.
- Determine the values of trigonometric ratios for complementary angles.
- Determine the values of trigonometric ratios for any angle using a calculator, a unit circle, corresponding reference angles, or special right-angled triangles.
The Four Quadrants and Trigonometric Reference Angles
Illustrate reference angles \(\alpha\) and the signs of trigonometric functions across all four quadrants (All, Sin, Tan, Cos).
Quadrant I
All ratios are positive. Reference angle: \(\alpha = \theta\).
Quadrant II
\(\sin \theta\) and \(\operatorname{cosec} \theta\) are positive. Reference angle: \(\alpha = 180^\circ - \theta\).
Quadrant III
\(\tan \theta\) and \(\cot \theta\) are positive. Reference angle: \(\alpha = \theta - 180^\circ\).
Quadrant IV
\(\cos \theta\) and \(\sec \theta\) are positive. Reference angle: \(\alpha = 360^\circ - \theta\).
Short Explanation
Use reciprocal definitions, complementary angle identities, reference angles and special triangles to evaluate any trigonometric ratio.
1. Reciprocal Trigonometric Ratios
\(\operatorname{cosec} \theta = \dfrac{1}{\sin \theta}\), \(\sec \theta = \dfrac{1}{\cos \theta}\), \(\cot \theta = \dfrac{1}{\tan \theta}\).
2. Complementary Angle Formulae
\(\sin \theta = \cos(90^\circ - \theta)\), \(\cos \theta = \sin(90^\circ - \theta)\), \(\tan \theta = \cot(90^\circ - \theta)\).
3. Unit Circle Method
For a point \((x,y)\) on a unit circle: \(\cos \theta = x\), \(\sin \theta = y\), \(\tan \theta = \dfrac{y}{x}\).
4. Reference Angle Method
Locate the quadrant, determine the sign, obtain the acute reference angle \(\alpha\), then use the ratio of \(\alpha\).
5. Special Right-Angled Triangles
Use \(30^\circ\), \(45^\circ\), and \(60^\circ\) triangles to find exact values such as \(\sin 30^\circ = \dfrac{1}{2}\).
6. Negative Angle Relationships
\(\sin(-A) = -\sin A\), \(\cos(-A) = \cos A\), \(\tan(-A) = -\tan A\).
| Quadrant | Positive Ratios | Reference Angle \(\alpha\) |
|---|---|---|
| I | All | \(\alpha = \theta\) |
| II | \(\sin, \operatorname{cosec}\) | \(\alpha = 180^\circ - \theta\) |
| III | \(\tan, \cot\) | \(\alpha = \theta - 180^\circ\) |
| IV | \(\cos, \sec\) | \(\alpha = 360^\circ - \theta\) |
Steps for Evaluating Trigonometric Ratios Without a Calculator
Locate the position of the angle in the quadrant.
Determine the sign for the trigonometric ratio.
Obtain the corresponding reference angle \(\alpha\).
Use the trigonometric ratio value of the reference angle.
Reciprocal ratios
\(\operatorname{cosec} \theta = \dfrac{1}{\sin \theta}, \quad \sec \theta = \dfrac{1}{\cos \theta}, \quad \cot \theta = \dfrac{1}{\tan \theta}\)Use these to evaluate reciprocal ratios from sine, cosine and tangent.
Complementary angle identities
\(\sin \theta = \cos(90^\circ - \theta), \quad \sec \theta = \operatorname{cosec}(90^\circ - \theta)\)A function of \(\theta\) equals the co-function of \(90^\circ - \theta\).
Try to Answer First
Answer in your mind, then press "Check Answer".
Given \(\cos 63^\circ = k\), find \(\operatorname{cosec} 27^\circ\) in terms of \(k\).
What is the reference angle \(\alpha\) for \(\theta = 240^\circ\)?
Find the exact value of \(\sec 150^\circ\) using reference angles.
Common Mistakes
- Confusing reciprocal pairs: thinking \(\operatorname{cosec} \theta = \dfrac{1}{\cos \theta}\) instead of \(\dfrac{1}{\sin \theta}\), and \(\sec \theta = \dfrac{1}{\sin \theta}\) instead of \(\dfrac{1}{\cos \theta}\).
- Calculating the reference angle with respect to the \(y\)-axis rather than always to the horizontal \(x\)-axis.
- Omitting the quadrant sign when converting from the original angle to the reference angle, for example writing \(\cos 120^\circ = \cos 60^\circ\) instead of \(-\cos 60^\circ\).
Concept Misunderstandings
Negative angles always yield negative trigonometric values.
\(\cos(-\theta) = \cos \theta\) and \(\sec(-\theta) = \sec \theta\) are positive if \(\theta\) is acute, because negative acute angles lie in Quadrant IV.
The reciprocal is the inverse function, for example \(\sec \theta = \cos^{-1} x\) or \(\arccos x\).
\(\sec \theta = \dfrac{1}{\cos \theta}\) is a reciprocal ratio, not the inverse trigonometric function \(\cos^{-1} x\).
Summary
- Reciprocal ratios: \(\operatorname{cosec} \theta = \dfrac{1}{\sin \theta}\), \(\sec \theta = \dfrac{1}{\cos \theta}\), \(\cot \theta = \dfrac{1}{\tan \theta}\).
- Complementary relationships: \(\sin \theta = \cos(90^\circ - \theta)\), \(\sec \theta = \operatorname{cosec}(90^\circ - \theta)\), \(\tan \theta = \cot(90^\circ - \theta)\).
- Reference angle \(\alpha\) is always acute and measured to the \(x\)-axis: Quadrant I \(\alpha = \theta\), Quadrant II \(\alpha = 180^\circ - \theta\), Quadrant III \(\alpha = \theta - 180^\circ\), Quadrant IV \(\alpha = 360^\circ - \theta\).
- Quadrant signs: Q1 (All positive), Q2 (\(\sin, \operatorname{cosec}\) positive), Q3 (\(\tan, \cot\) positive), Q4 (\(\cos, \sec\) positive).
Short Activity
Objective: Identify reciprocal ratios, complementary angle identities, reference angles and special angle values.
A. Objective Quiz
1 Given that \(\theta\) lies in Quadrant III, which pair of trigonometric ratios are positive?
2 Which of the following expressions is equivalent to \(\sec(90^\circ - \theta)\)?
B. Fill in the Blanks
3 For an angle \(\theta = 315^\circ\), the corresponding acute reference angle \(\alpha\) is ________ degrees.
4 The reciprocal ratio of \(\cos \theta\) is ________.
C. Matching / Drag and Drop
Drag each expression to the correct identity or description. On a phone, tap an expression first, then tap the matching box.
