# Trigonometry Chapters 8.2 - 8.3. 45-45-90 Theorem.

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Trigonometry Chapters 8.2 - 8.3

45-45-90 Theorem

The opposite sides of a 45-45-90 triangle are the same length Using the Pythagorean theorem we find the hypotenuse is always n times the square root of 2

45-45-90 Theorem What is the length of the hypotenuse

45-45-90 Theorem What is the length of the hypotenuse

45-45-90 Theorem What is the length of the sides?

45-45-90 Theorem What is the length of the sides? Remember, the hypotenuse is  2 times a side

Divide by  2 Rationalize the Denominator

30-60-90 Theorem

The opposite of the 30 0 angle is n

30-60-90 Theorem The opposite of the 60 0 angle is n  3

30-60-90 Theorem The opposite of the right angle is 2n

30-60-90 Theorem Find the lengths of the other two sides

30-60-90 Theorem Find the lengths of the other two sides

30-60-90 Theorem Find the lengths of the other two sides

30-60-90 Theorem Find the lengths of the other two sides

30-60-90 Theorem Find the lengths of the other two sides First find the length of side opposite the 30

30-60-90 Theorem Call the side x x times  3 = 8

30-60-90 Theorem Hypotenuse is 2 times the side opposite the 30 0 angle

Trigonometry Trigonometric Ratios- – Similar right triangles have equivalent ratios for its corresponding sides

Sine Sine of óB =

Sine Sine of óB =

Sine Sin B =

Cosine Cosine of óB =

Cosine Cosine of óB =

Cosine Cos B =

Tangent Tangent of óB =

Tangent Tangent of óB =

Tangent Tan B =

Trigonometry How to remember the order: Sin x = Cos x = Tan x =

Trigonometry Find the sine, cosine, and tangent ratios of ó B

Sin B = Cos B = Tan B =

Trigonometry What is the sin of ó B? Type it into a calculator: sin (40) Sin (40) =.64

How can we use this? Find the length of the hypotenuse. We’re given angle B and the opposite side

We know: We can plug in what we know: Find the length of the hypotenuse. Sin B = Sin 40 =

Solve: Sin 40 = Sin 40 =

Plug it into your calculator to find x Sin 40 = =

Find the length of the adjacent side. Cos B = Cos 40 =

Solve for x Cos 40 = Type into calculator 9.19 =

Find the length of the adjacent side. 9.19 =

Inverse Trig Functions Each trig function has an inverse that works like dividing. The inverse of sin is sin -1

Inverse Trig Functions

Find angle x Label the triangle

We have the opposite and hypotenuse Which one uses those two sides? Sin x = Cos x = Tan B =

Plug in the opposite and hypotenuse Sin x = Notice the x is in front of the sine, so we can’t just divide! Multiply by the inverse of sine

This cancels out the sine on the left Sin -1 (Sin x) = Sin -1