22Aexactvals

= Exact Values =

When we calculate the sin, cos or tan of most angles, we have to use approximate answers (decimals).

For a small number of angles, we can write the exact answers.

The angles we have exact values for are: 0º, 30º, 45º, 60º, 90º.

0º
Recall that on the unit circle, y = sin(q ), x = cos(q ) math . \qquad \text{and } \quad \tan \big( \theta \big) = \dfrac{ \sin \big( \theta \big) }{ \cos \big( \theta \big) } math

Hence:

math \\ . \qquad \bullet \quad \sin \big( 0^\circ \big) = 0 \\. \\ . \qquad \bullet \quad \cos \big( 0^\circ \big) = 1 \\. \\ . \qquad \bullet \quad \tan \big( 0^\circ \big) = 0 math

90º
Recall that on the unit circle, y = sin(q ), x = cos(q ) math . \qquad \text{and } \quad \tan \big( \theta \big) = \dfrac{ \sin \big( \theta \big) }{ \cos \big( \theta \big) } math

Hence: math \\ . \qquad \bullet \quad \sin \big( 90^\circ \big) = 1 \\. \\ . \qquad \bullet \quad \cos \big( 90^\circ \big) = 0 \\. \\ . \qquad \bullet \quad \tan \big( 90^\circ \big) = \text{undefined} math

30º
math . \qquad \bullet \quad \sin \big( 30^\circ \big) = \dfrac{1}{2} math

If we know that sin(30º) = ½, then we can construct the triangle shown here:

Use Pythagoras to find the missing side:

math \\ . \qquad a^2 = c^2 - b^2 \\. \\ . \qquad \quad = 2^2 - 1^1 \\. \\ . \qquad \quad = 4 - 1 \\. \\ . \qquad \quad = 3 math

{take the square root of both sides}

math . \qquad a = \sqrt{3} math

Now we can improve the triangle we drew earlier.

Hence, using SOHCAHTOA, we can find the trig values for 30º.

math \\ . \qquad \bullet \quad \sin \big( 30^\circ \big) = \dfrac{1}{2} \\. \\ . \qquad \bullet \quad \cos \big( 30^\circ \big) = \dfrac{\sqrt{3}}{2} \\. \\ . \qquad \bullet \quad \tan \big( 30^\circ \big) = \dfrac{1}{\sqrt{3}} math

60º
We can improve the above triangle by noting that the missing angle is 60º. {the sum of angles in a triangle is 180º}

Hence, using SOHCAHTOA, we can find the trig values for 60º.

math \\ . \qquad \bullet \quad \sin \big( 60^\circ \big) = \dfrac{\sqrt{3}}{2} \\. \\ . \qquad \bullet \quad \cos \big( 60^\circ \big) = \dfrac{1}{2} \\. \\ . \qquad \bullet \quad \tan \big( 60^\circ \big) = \dfrac{\sqrt{3}}{1}=\sqrt{3} math

45º
math . \qquad \bullet \quad \tan \big( 45^\circ \big) = 1 math

If we know that tan(45º) = 1, we can construct the triangle shown here:

Use Pythagoras to find the hypotenuse:

math \\ . \qquad c^2 = a^2 + b^2 \\. \\ . \qquad \quad = 1^2 + 1^2 \\. \\ . \qquad \quad = 1 + 1 \\. \\ . \qquad \quad = 2 math

{take the square root of both sides}

math . \qquad c = \sqrt{2} math

Now we can improve the triangle we drew earlier.

Hence, using SOHCAHTOA, we can find the trig values for 45º.

math \\ . \qquad \bullet \quad \sin \big( 45^\circ \big) = \dfrac{1}{\sqrt{2}} \\. \\ . \qquad \bullet \quad \cos \big( 45^\circ \big) = \dfrac{1}{\sqrt{2}} \\. \\ . \qquad \bullet \quad \tan \big( 45^\circ \big) =1 math

Using Exact Values
Knowing these exact values allows us to do a limited amount of trigonometry without a calculator.

All we need to remember are two facts:

math \\ . \qquad \bullet \quad \sin \big( 30^\circ \big) = \dfrac{1}{2} \\. \\ . \qquad \bullet \quad \tan \big( 45^\circ \big) = 1 math

From these two facts, we can produce the two triangles: ...

Then we can use SOHCAHTOA to produce the harder parts of the following table:



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