Question

For the angles α and β in the figures, find cos(α + β)?

Answer 1

`cos(\alpha + \beta)=cos(68.4\°) \approx 0.37`

Explanation:

Little triangle.

We know both legs, we can use the tangent trigonometric reason to find the angle.

`tan\alpha =(2)/(4)\\ tan \alpha=(1)/(2)\\ \alpha=tan^(-1)((1)/(2) )\\ \alpha \approx 26.6\°`

Larger triangle.

We know the hypothenuse and the opposite leg. We can use the sin trigonometric reason to find the angle

`sin\beta =(4)/(6)\\ sin\beta=(2)/(3)\\ \beta=sin^(-1) ((2)/(3) )\\\beta= 41.8\°`

So, the sum of them is

`\alpha + \beta = 26.6+41.8= 68.4\°`

Then,

`cos(\alpha + \beta)=cos(68.4\°) \approx 0.37`

Therefore,

`cos(\alpha + \beta)=cos(68.4\°) \approx 0.37`

Answer 2

`\cos(\alpha +\beta)=(2)/(3)(1-(√(5))/(5))`

Explanation:

Let the hypotenuse of the smaller triangle be h units.

Then; from the Pythagoras Theorem.

`h^2=4^2+2^2`

`h^2=16+4`

`h^2=20`

`h=√(20)`

`h=2√(5)`

From the smaller triangle;

`\cos (\alpha)=(4)/(2√(5) )=(2)/(√(5) )` and
`\sin(\alpha)=(2)/(2√(5) )=(1)/(√(5) )`

From the second triangle, let the other other shorter leg of the second triangle be s units.

Then;

`s^2+4^2=6^2`

`s^2+16=36`

`s^2=36-16`

`s^2=20`

`s=√(20)`

`s=2√(5)`

`\cos(\beta)=(2√(5) )/(6)=(√(5) )/(3)`

and

`\sin(\beta)=(4)/(6)=(2)/(3)`

We now use the double angle property;

`\cos(\alpha +\beta)=\cos(\alpha)\cos(\beta) -\sin(\alpha)\sin(\beta)`

we plug in the values to obtain;

`\cos(\alpha +\beta)=(2)/(√(5) )* (√(5) )/(3)-(1)/(√(5) )* (2)/(3)`

`\cos(\alpha +\beta)=(2)/(3)(1-(√(5))/(5))`