What angles can I use in trigonometry

The right triangle

Designations in right-angled triangle

The longest Page in right-angled The triangle is opposite the right angle. her name is hypotenuse.

The other two sides are called Catheters.

Opposite and adjacent

The Catheters are distinguished again.

The leg that corresponds to the angle $$ alpha $$ againstabove is called Againstcathete from $$ alpha $$.

The cathetus that is at the angle $$ alpha $$ atlies, is called Atcathete from $$ alpha $$.

Example:

Page $$ a $$:
Since the side $$ a $$ corresponds to the angle $$ alpha $$ againstthe side $$ a $$ is the AgainstCathete of the angle $$ alpha $$. Since the side $$ a $$ also at the angle $$ beta $$ atit is at the same time that Atcathete from $$ beta $$.

Page $$ b $$:
Since the side $$ b $$ corresponds to the angle $$ beta $$ againstthe side $$ b $$ is the Againstcathete of the angle $$ beta $$. Since the side $$ b $$ also at the angle $$ alpha $$ atit is at the same time that Atcathete from $$ alpha $$.

trigonometry

Now is the time to calculate. The part of math that calculates sides and angles in triangles is called trigonometry. Let's go with right-angled Triangles.

In right triangles you can discover the same length ratios.

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The sine of an angle

a) $$ alpha = 30 ° $$; $$ a = 2 \ cm $$; $$ c = 4 \ cm $$

b) $$ α = 30 ° $$; $$ a = 3 \ cm $$; $$ c = 6 \ cm $$

The quotient $$ a / c = (opposite \ enkathete) / (hypoten \ use) $$ has for both right-angled Triangles have the same value.

a) $$ a / c = 2/4 = 1/2 $$

b) $$ a / c = 3/6 = 1/2 $$

This aspect ratio becomes Sine called.

In the right triangle the following applies:

$$ S \ i \ n \ us = (Opposite \ enkathete) / (Hypoten \ use) $$

The cosine of an angle

The quotient $$ b / c = (adjacent) / (hypoten \ use) $$ has both right-angled Triangles have the same value.

This aspect ratio becomes cosine called.

In the right triangle the following applies:

$$ K \ o \ si \ n \ us = (adjacent) / (hypoten \ use) $$

The tangent of an angle

The quotient $$ a / b = (Ge \ g \ e \ nkathete) / (adjacent) $$ has for both right-angled Triangles have the same value.

This aspect ratio becomes tangent called.

In the right triangle the following applies:

$$ Tang \ ens = (Ge \ g \ e \ nkathete) / (adjacent) $$

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Simple calculations with the trigonometric functions

Example 1: calculate pages

given: $$ c = 4 \ cm $$; $$ alpha = 30 ° $$; $$ gamma = 90 ° $$

Page $$ a $$

1. Set up a formula

$$ sin alpha = (opposite \ enkathete) / (hypoten \ use) $$ $$ | * c $$


2. Change the formula

$$ sin alpha = (Opposite \ enkathete) / (Hypoten \ use) $$ $$ | * c $$
$$ c * sin alpha = a $$


3. Calculate

$$ 4 * sin 30 ° = a $$
$$ 2 \ cm = a $$

Side b

1. Set up a formula

$$ cos β = (adjacent) / (hypoten \ use) $$ $$ | * c $$

2. Change the formula

$$ cos β = (adjacent) / (hypoten \ use) $$ $$ | * c $$
$$ c * cos β = b $$

3. Calculate

$$ 4 * cos 30 ° = b $$
$$ 3.46 cm ≈ b $$

TR input:
  $$4$$  
  $$*$$  
TR input:
  $$4$$  
  $$*$$  

Simple calculations with the trigonometric functions

Example 2: calculate angle

$$ a = 3 \ cm $$; $$ b = 4 \ cm $$; $$ alpha =? $$

Angle $$ alpha $$

1. Set up a formula

$$ tan alpha = (opposite side) / (adjacent side) = a / b $$

2. Calculate

$$ tan alpha = 3/4 $$

$$ alpha ≈ 36.87 ° $$

TR input:
  $$3/4$$