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.
Illustrate reference angles \(\alpha\) and the signs of trigonometric functions across all four quadrants (All, Sin, Tan, Cos).
All ratios are positive. Reference angle: \(\alpha = \theta\).
\(\sin \theta\) and \(\operatorname{cosec} \theta\) are positive. Reference angle: \(\alpha = 180^\circ - \theta\).
\(\tan \theta\) and \(\cot \theta\) are positive. Reference angle: \(\alpha = \theta - 180^\circ\).
\(\cos \theta\) and \(\sec \theta\) are positive. Reference angle: \(\alpha = 360^\circ - \theta\).
Use reciprocal definitions, complementary angle identities, reference angles and special triangles to evaluate any trigonometric ratio.
\(\operatorname{cosec} \theta = \dfrac{1}{\sin \theta}\), \(\sec \theta = \dfrac{1}{\cos \theta}\), \(\cot \theta = \dfrac{1}{\tan \theta}\).
\(\sin \theta = \cos(90^\circ - \theta)\), \(\cos \theta = \sin(90^\circ - \theta)\), \(\tan \theta = \cot(90^\circ - \theta)\).
For a point \((x,y)\) on a unit circle: \(\cos \theta = x\), \(\sin \theta = y\), \(\tan \theta = \dfrac{y}{x}\).
Locate the quadrant, determine the sign, obtain the acute reference angle \(\alpha\), then use the ratio of \(\alpha\).
Use \(30^\circ\), \(45^\circ\), and \(60^\circ\) triangles to find exact values such as \(\sin 30^\circ = \dfrac{1}{2}\).
\(\sin(-A) = -\sin A\), \(\cos(-A) = \cos A\), \(\tan(-A) = -\tan A\).
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
Use these to evaluate reciprocal ratios from sine, cosine and tangent.
Complementary angle identities
A function of \(\theta\) equals the co-function of \(90^\circ - \theta\).
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.
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 \(\cos^{-1} x\).
Objective: Identify reciprocal ratios, complementary angle identities, reference angles and special angle values.
1 Given that \(\theta\) lies in Quadrant III, which pair of trigonometric ratios are positive?
2 Which of the following expressions is equivalent to \(\sec(90^\circ - \theta)\)?
3 For an angle \(\theta = 315^\circ\), the corresponding acute reference angle \(\alpha\) is ________ degrees.
4 The reciprocal ratio of \(\cos \theta\) is ________.
Drag each expression to the correct identity or description. On a phone, tap an expression first, then tap the matching box.
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