UPSKILL MATH PLUS
Learn Mathematics through our AI based learning portal with the support of our Academic Experts!
Learn moreConsider an isosceles triangle ABC with angle measure 45^{\circ} right-angled at B.
Let the measure of the equal sides of the triangle be a units.
First, let us calculate the measure of the hypotenuse in the figure.
Since the given triangle is a right-angled triangle by the Pythagoras theorem, we have:
In a right angled triangle, \text{Hypotenuse}^{2} = \text{Adjacent side}^{2} + \text{Opposite side}^{2}.
AC^2 = AB^2 + BC^2.
AC^2 = a^2 + a^2
AC^2 = 2a^2
\Rightarrow AC = \sqrt{2a^2}
AC = a \sqrt{2} units
Therefore, for the given right-angled isosceles triangle we have:
Opposite side = a units
Adjacent side = a units
Hypotenuse = a \sqrt{2} units
Let us determine all the trigonometric ratios of 45^{\circ}.
- Sine 45^{\circ}:
\sin 45^{\circ} = \frac{\text{Opposite side}}{\text{Hypotenuse}}
= \frac{a}{a \sqrt{2}}
= \frac{1}{\sqrt{2}}
- Cosine 45^{\circ}:
\cos 45^{\circ} = \frac{\text{Adjacent side}}{\text{Hypotenuse}}
= \frac{a}{a \sqrt{2}}
= \frac{1}{\sqrt{2}}
- Tangent 45^{\circ}:
\tan 45^{\circ} = \frac{\text{Opposite side}}{\text{Adjacent side}}
= \frac{a}{a}
= 1
Using these basic trigonometric ratios determine their reciprocals as follows:
- Cosecant 45^{\circ}:
\text{cosec}\,\theta = \frac{1}{\sin \theta}
= \sqrt{2}
- Secant 45^{\circ}:
\sec \theta = \frac{1}{\cos \theta}
= \sqrt{2}
- Cotangent 45^{\circ}:
\cot \theta = \frac{1}{\tan \theta}
= 1
Let us summarize all the trigonometric ratios of 45^{\circ} in the following table.
\sin \theta | \cos \theta | \tan \theta | \text{cosec}\,\theta | \sec \theta | \cot \theta | |
\theta = 45^{\circ} | \frac{1}{\sqrt{2}} | \frac{1}{\sqrt{2}} | 1 | \sqrt{2} | \sqrt{2} | 1 |
Register for free to see more content