What is the correct definition for tan 0

Mathematics

1 answer:

Reptile [31]2 years ago

3 0

$tan\theta=\frac{sin\theta}{cos\theta}\\\\\\Look\ at\ the\ picture.\\\\In\ a\ right\ triangle:\\\\sin\theta=\frac{a}{c};\ cos\theta=\frac{b}{c};\ tan\theta=\frac{a}{b}\\\\\frac{sin\theta}{cos\theta}=\frac{\frac{a}{c}}{\frac{b}{c}}=\frac{a}{c}\times\frac{c}{b}=\frac{a}{b}=tan\theta$

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Dvinal [7]

Step-by-step explanation:

Since m∠ACM = 90°, then by angle addition, m∠ACP + m∠PCM = 90°.
Since CP⊥ AM, then by definition of perpendicular, m∠APC = 90°. Since angles of a triangle add up to 180°, that means m∠PAC + m∠ACP = 90°.
By substitution, m∠PCM = m∠PAC.
Since m∠PCM + m∠CMP = 90°, then by substitution, m∠CMP = m∠PAC.
Therefore, by AAA, △ACP and △CMP are similar.

Having proven that the triangles are similar, we can write the proportion:

AP / CP = CP / MP

9 / CP = CP / 16

CP² = 144

CP = 12

Now, we can simply use Pythagorean theorem to find the other sides.

AC² = AP² + CP²

AC² = 9² + 12²

AC = 15

CM² = CP² + PM²

CM² = 12² + 16²

CM = 20