Answer to Question #106089 in Geometry for Raheem

Question

Another student has heard how smart you are and needs help finding the measure of 2 angles after being told only that Sine = Cos and that angle has (4x + 10) degrees and α β α that angle has (10x + 10) degrees. Please help him to find the measure of each angle and β add an explanation of the rules applied here and the steps taken so that this student can do this on their own next time.

Answer 1

It is stated in the task sin=cos

We do not know if it means sin$\alpha$ = cos $\alpha$ , sin $\beta$ = cos $\beta$ , sin $\alpha$ = cos $\beta$ , or sin $\beta =$ cos $\alpha$

1) Let us look at the last 2 cases (sin $\alpha$ = cos $\beta$ , or sin $\beta$ = cos $\alpha$ )

sin $\alpha$ = cos $\beta$ $\rArr \alpha=90^o-\beta \rArr$ sin $\beta$ = cos $\alpha$

So this cases are actually the same

We know that $\alpha=4x+10$ and $\beta = 10x + 10$

We can substitute these into $\alpha=90^o-\beta$

$\alpha=90^o-\beta \rArr 4x+10=90-(10x+10) \rArr 4x+10=90-10x-10$

$(4x+10)+10x-10=(90-10x-10)+10x-10$

$14x=70 \rArr x=5 \rArr \alpha = 4*5+10=30^o \rArr \beta=90^o-\alpha=90^o-30^o=60^o$

2) Let us look at the case sin $\alpha$ = cos $\alpha$ .

This means that $\alpha=45^o \rArr 4x+10=45^o \rArr 4x=35^o \rArr 10x=35*\frac{10}{4}=87^o.5$

$\beta=10x+10=97^o.5$

3) Let us look at the case sin $\beta =$ cos $\beta$

This means that $\beta = 45^o \rArr 10x+10=45^o \rArr 10x=35^o \rArr 4x=35^o\frac{4}{10}=14^o$

$\alpha=4x+10=24^o$