30. Five mark exercise problems VI

1. From the given figure, prove that $\theta + \phi = 90^{\circ}$. Also prove that there are two other right angled triangles. Find $sin \ \alpha$, $cos \ \beta$ and $tan \ \phi$.

 

 

Answer:

 

$sin \ \alpha =$ $\frac{i}{i}$

 

$cos \ \beta =$ $\frac{i}{i}$

 

$tan \ \phi =$ $\frac{i}{i}$

 

2. Verify $cos \ 90^{\circ} = 1 - 2 \ sin^2 \ 45^{\circ} = 2 \ cos^2 \ 45^{\circ} - 1$

 

Proof:

 

$cos \ 90^{\circ} =$

 

$1 - 2 \ sin^2 \ 45^{\circ} =$

 

$2 \ cos^2 \ 45^{\circ} - 1 =$

 

Thus, $cos \ 90^{\circ} = 1 - 2 \ sin^2 \ 45^{\circ} = 2 \ cos^2 \ 45^{\circ} - 1$.

 

Hence, we proved.