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**Answers to math exercises in 1st grade on trigonometry**.

Exercise 1:

Solve in the following equations.

1.

2.

Exercise 2:

In this exercise, we give:

Calculate the exact value of and then of

so

** Hint: ** for , use the addition formula with and .

Exercise 3:

In this exercise, we have the following data:

1. Let . Prove that

2. Deduce that :

Use the equality of 1. by taking

so

and using the data from the statement :

Let’s multiply by the conjugate quantity and use the remarkable identity .

**Conclusion:**

Exercise 4:

**Solve in the equation: sin(2x) = cos(x).**

It’s a product equation.

A product of factors is zero if and only if at least one of the factors is zero.

or

Exercise 5:

Solve in the following equations:

Hint: make changes of variable and use trigonometric formulas.

For the 1, put X=cosx.

Exercise 6:

Solve in the equation :

Let’s put

We have to solve :

An obvious root is X = 1.

a = 2

-c=8 so c= -8

c-b=7 so -8-b=7 so b=-8-7=-15

So

Let’s calculate the value of the discriminant :

The discriminant is positive, there are two distinct real roots.

so

so we get :

or

so either

Conclusion:

Exercise 7:

Exercise 8:

Exercise 9:

Exercise 10:

ABC is a triangle with .

1. Prove that .

2. Calculate the exact values of AB and AC .

Indications:

Use the Al-Kashi formulas and the generalized Pythagorean theorem.

Exercise 11:

Show that the graphical representation of the function defined on by :

is located between the lines of equation y = – 3 and y = 1 .

All this comes from the fact that and .

Just add up each member.

Exercise 12:

Show that, for any real :

Let’s use the remarkable identity

because and

Conclusion:

Exercise 13:

Using the addition formulas, calculate the exact value of

Exercise 14:

Let ABC be any triangle.

In the triangles AEB and BEC rectangles in E ,

by applying the Pythagorean theorem :

so by equality, we deduce that :

** (*) **

In the triangle AEB right-angled at E, using right-angled trigonometry:

so

Let’s take the equality ( ** *) **

**Remark: ** these are the same demonstrations for the other formulas of Al-Kashi,

which are also called generalized Pythagorean formulas.

Exercise 15:

Exercise 16:

Exercise 17:

## Answers to the exercises on trigonometry in 1st grade.

After having consulted the **answers to** these **exercises** on trigonometry in 1ère, you can return to the **exercises** in **première**.

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