In triangle ABC, right-angled at B , AB =5, cm and ∠ ACB =30^{circ} (see Fig. ). Determine lengths

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8. Introduction to Trigonometry

hard

In $\triangle$ $ABC,$ right-angled at $B$, $AB =5\, cm$ and $\angle ACB =30^{\circ}$ (see $Fig.$). Determine the lengths of the sides $BC$ and $AC .$

Option A

Option B

Option C

Option D

Solution

To find the length of the side $BC ,$ we will choose the trigonometric ratio involving $BC$ and the given side $AB$. since $BC$ is the side adjacent to angle $C$ and $AB$ is the side opposite to angle $C ,$ therefore

$\frac{ AB }{ BC }=\tan C$

$\frac{5}{ BC }=\tan 30^{\circ}=\frac{1}{\sqrt{3}}$

which gives $BC =5 \sqrt{3} \,cm$

To find the length of the side $AC ,$ we consider

$\sin 30^{\circ}=\frac{ AB }{ AC }$

$\frac{1}{2}=\frac{5}{ AC }$

$AC =10 \,cm$

Note that alternatively we could have used Pythagoras theorem to determine the third side in the example above,

$AC =\sqrt{ AB ^{2}+ BC ^{2}}=\sqrt{5^{2}+(5 \sqrt{3})^{2}} cm =10 \,cm$

Standard 10

Mathematics

If $\sin 3 A =\cos \left( A -26^{\circ}\right),$ where $3 A$ is an acute angle, find the value of $A= . . . . ^{\circ}$.

medium

Prove that

$\frac{\sin \theta-\cos \theta+1}{\sin \theta+\cos \theta-1}=\frac{1}{\sec \theta-\tan \theta},$ using the identity

$\sec ^{2} \theta=1+\tan ^{2} \theta$

hard

Evaluate $\frac{\tan 65^{\circ}}{\cot 25^{\circ}}$

easy

Consider $\triangle ACB$, right-angled at $C$, in which $AB =29$ units, $BC =21$ units and $\angle ABC =\theta$ (see $Fig.$). Determine the values of

$(i)$ $\cos ^{2} \theta+\sin ^{2} \theta$

$(ii)$ $\cos ^{2} \theta-\sin ^{2} \theta$

medium

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

$\frac{1+\sec A}{\sec A}=\frac{\sin ^{2} A}{1-\cos A}$

medium

In $\triangle$ $ABC,$ right-angled at $B$, $AB =5\, cm$ and $\angle ACB =30^{\circ}$ (see $Fig.$). Determine the lengths of the sides $BC$ and $AC .$

Solution

Similar Questions

If $\sin 3 A =\cos \left( A -26^{\circ}\right),$ where $3 A$ is an acute angle, find the value of $A= . . . . ^{\circ}$.

Prove that $\frac{\sin \theta-\cos \theta+1}{\sin \theta+\cos \theta-1}=\frac{1}{\sec \theta-\tan \theta},$ using the identity $\sec ^{2} \theta=1+\tan ^{2} \theta$

Evaluate $\frac{\tan 65^{\circ}}{\cot 25^{\circ}}$

Consider $\triangle ACB$, right-angled at $C$, in which $AB =29$ units, $BC =21$ units and $\angle ABC =\theta$ (see $Fig.$). Determine the values of $(i)$ $\cos ^{2} \theta+\sin ^{2} \theta$ $(ii)$ $\cos ^{2} \theta-\sin ^{2} \theta$

Prove the following identities, where the angles involved are acute angles for which the expressions are defined. $\frac{1+\sec A}{\sec A}=\frac{\sin ^{2} A}{1-\cos A}$

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Prove that

$\frac{\sin \theta-\cos \theta+1}{\sin \theta+\cos \theta-1}=\frac{1}{\sec \theta-\tan \theta},$ using the identity

$\sec ^{2} \theta=1+\tan ^{2} \theta$

Consider $\triangle ACB$, right-angled at $C$, in which $AB =29$ units, $BC =21$ units and $\angle ABC =\theta$ (see $Fig.$). Determine the values of

$(i)$ $\cos ^{2} \theta+\sin ^{2} \theta$

$(ii)$ $\cos ^{2} \theta-\sin ^{2} \theta$

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

$\frac{1+\sec A}{\sec A}=\frac{\sin ^{2} A}{1-\cos A}$