\(\text{cosec }\theta= \dfrac{1}{\text{sin }\theta}\)

\(\text{sec }\theta= \dfrac{1}{\text{cos }\theta}\)

\(\text{cot }\theta= \dfrac{1}{\text{tan }\theta}\)

The angles \(A\) and \(B\) are complementary angles to each other if \(A+B=90^\circ\). Hence,

\(A=90^\circ-B\) and \(B=90^\circ-A\)

• \(\text{sin }\theta= \text{cos } (90^{\circ}-\theta)\)
• \(\text{cos }\theta= \text{sin } (90^{\circ}-\theta)\)
• \(\text{tan }\theta= \text{cot } (90^{\circ}-\theta)\)
• \(\text{sec }\theta= \text{cosec } (90^{\circ}-\theta)\)
• \(\text{cosec }\theta= \text{sec } (90^{\circ}-\theta)\)
• \(\text{cot }\theta= \text{tan } (90^{\circ}-\theta)\)

• The values of sine, cosine and tangent can be determined by using a calculator.
• The values for cosecant, secant and cotangent can be calculated by inversing the values of the trigonometric ratios of sine, cosine and tangent of that particular angle.

The diagram shows the reference angles, \(\alpha\) for the angles \( 0° \leqslant \theta \leqslant 360^{\circ}\) or \( 0° \leqslant \theta \leqslant 2\pi\).

The trigonometric ratios of special angles \(30^\circ\), \(45^\circ\) and \(60^\circ\) can be determined by using right-angled triangles.

(a)

\(\begin{aligned} \text{cos }\theta&= \text{sin } (90^{\circ}-\theta)\\ \text{cos }13^{\circ}&= \text{sin } (90^{\circ}-13^{\circ})\\ &= \text{sin }77^{\circ}\\ &= 0.9744. \end{aligned}\)

(b)

\(\begin{aligned} \text{cosec }\theta&= \text{sec } (90^{\circ}-\theta)\\ \text{cosec }27^{\circ}&= \text{sec } (90^{\circ}-27^{\circ})\\ &= \text{sec }63^{\circ}\\ &= \dfrac{1}{\text{cos }63^{\circ}}\\ &= \dfrac{1}{k}. \end{aligned}\)

Use the unit circle and state the value of \(\text{cos }135^{\circ}\).

The coordinates that correspond to \(135^{\circ}\) are

\(\begin{pmatrix} -\dfrac{1}{\sqrt2}, \ \dfrac{1}{\sqrt2} \end{pmatrix}\) and \(\text{cos }135^{\circ} = x\text{-coordinate}\).

Hence, \(\text{cos }135^{\circ} = -\dfrac{1}{\sqrt2}\).

\(\text{cosec }\theta= \dfrac{1}{\text{sin }\theta}\)

\(\text{sec }\theta= \dfrac{1}{\text{cos }\theta}\)

\(\text{cot }\theta= \dfrac{1}{\text{tan }\theta}\)

The angles \(A\) and \(B\) are complementary angles to each other if \(A+B=90^\circ\). Hence,

\(A=90^\circ-B\) and \(B=90^\circ-A\)

• \(\text{sin }\theta= \text{cos } (90^{\circ}-\theta)\)
• \(\text{cos }\theta= \text{sin } (90^{\circ}-\theta)\)
• \(\text{tan }\theta= \text{cot } (90^{\circ}-\theta)\)
• \(\text{sec }\theta= \text{cosec } (90^{\circ}-\theta)\)
• \(\text{cosec }\theta= \text{sec } (90^{\circ}-\theta)\)
• \(\text{cot }\theta= \text{tan } (90^{\circ}-\theta)\)

• The values of sine, cosine and tangent can be determined by using a calculator.
• The values for cosecant, secant and cotangent can be calculated by inversing the values of the trigonometric ratios of sine, cosine and tangent of that particular angle.

The diagram shows the reference angles, \(\alpha\) for the angles \( 0° \leqslant \theta \leqslant 360^{\circ}\) or \( 0° \leqslant \theta \leqslant 2\pi\).

The trigonometric ratios of special angles \(30^\circ\), \(45^\circ\) and \(60^\circ\) can be determined by using right-angled triangles.

(a)

\(\begin{aligned} \text{cos }\theta&= \text{sin } (90^{\circ}-\theta)\\ \text{cos }13^{\circ}&= \text{sin } (90^{\circ}-13^{\circ})\\ &= \text{sin }77^{\circ}\\ &= 0.9744. \end{aligned}\)

(b)

\(\begin{aligned} \text{cosec }\theta&= \text{sec } (90^{\circ}-\theta)\\ \text{cosec }27^{\circ}&= \text{sec } (90^{\circ}-27^{\circ})\\ &= \text{sec }63^{\circ}\\ &= \dfrac{1}{\text{cos }63^{\circ}}\\ &= \dfrac{1}{k}. \end{aligned}\)

Use the unit circle and state the value of \(\text{cos }135^{\circ}\).

The coordinates that correspond to \(135^{\circ}\) are

\(\begin{pmatrix} -\dfrac{1}{\sqrt2}, \ \dfrac{1}{\sqrt2} \end{pmatrix}\) and \(\text{cos }135^{\circ} = x\text{-coordinate}\).

Hence, \(\text{cos }135^{\circ} = -\dfrac{1}{\sqrt2}\).