 This trigonometry course on cosine, sine and tangent of an acute angle and the calculation of the measure of an angle in a right triangle in 3rd grade is very instructive. In this lesson we will see how to calculate the length of a side of a right triangle knowing its cosine, sine or tangent.

Then, we will calculate the measure of one of the acute angles of a right-angled triangle knowing two of its lengths as well as using the calculator with the degree mode (DEG).

## I. Cosine, sine and tangent :

### 1 Definitions :

Definition:

Consider a right triangle and its acute angles.

1. The cosine of an acute angle is equal to the quotient of the side adjacent to the angle and thehypotenuse of the right triangle.
2. The sine of an acute angle is equal to the quotient of the side opposite the angle and the hypotenuse of the right triangle.
3. The tangent of an acute angle is equal to the quotient of the side opposite the angle and the side adjacent to the angle.

Example: Remark:

1. To remember these formulas easily, you can use the following mnemonic: SOH-CAH-TOA which corresponds to the initials in bold in the previous formulas.
2. The cosine and sine of an acute angle are always between 0 and 1.
3. The tangent of an acute angle is a strictly positive number.

### 2.Applications:

Example: In the right triangle LOE, calculate OE.

We know:

1. the angle ;
2. the length OL(hypotenuse)

We are looking for:

the length OE(opposite)

Formula:

The only formula that brings up the words “hypotenuse” and “opposite” is sine.  Using the cross product property: ## II. Calculating the measure of an angle :

Method:

To calculate the measure of an angle, make sure your calculator is in DEGRE mode.

The keys arccos, arcsin, arctan are used.

Example:

In the right triangle FUN, determine the measure of the angle rounded to the degree. In this right-angled triangle,

we know :

2. the length FU (opposite).

The only formula that brings up the words “adjacent” and “opposite” is the tangent.  We use the calculator in DEGRE mode and the “arctan” key. .

## Have you assimilated this course on trigonometry in the right-angled triangle in 3rd grade?

La trigonométrie dans le triangle rectangle

Un QCM sur la trigonométrie dans le triangle rectangle.

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