# 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 $$ alpha $$ **against**above is called **Against**cathete from $$ alpha $$.

The cathetus that is at the angle $$ alpha $$ **at**lies, is called **At**cathete from $$ alpha $$.

### Example:

**Page $$ a $$:**

Since the side $$ a $$ corresponds to the angle $$ alpha $$ **against**the side $$ a $$ is the **Against**Cathete of the angle $$ alpha $$. Since the side $$ a $$ also at the angle $$ beta $$ **at**it is at the same time that **At**cathete from $$ beta $$.

**Page $$ b $$:**

Since the side $$ b $$ corresponds to the angle $$ beta $$ **against**the side $$ b $$ is the **Against**cathete of the angle $$ beta $$. Since the side $$ b $$ also at the angle $$ alpha $$ **at**it is at the same time that **At**cathete 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:

##### TR input:

### 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**

$$ tan alpha = 3/4 $$

$$ alpha ≈ 36.87 ° $$

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