Trigonometry 1st S : corrected math exercises in PDF .

A series of corrected math exercises on trigonometry in 1st S is always beneficial. In addition, this chapter provides an opportunity to develop new skills. Trigonometry in 1st grade allows you to progress throughout the school year.

This sheet involves the following concepts:

addition formula;
trigonometry formulas;
trigonometric circle;
formulas of Al-Kashi;
generalized Pythagorean formula;
principal measure of an angle.

Exercise 1:

Let g be the function defined on $\mathbb{R}$ by :

1)Show that g is even. Interpret graphically.

2)Show that g is $\frac{\pi}{2}$ – periodic.

Exercise 2:

let g be the function defined on $\mathbb{R}$ by :

1)Show that g is neither even nor odd.

2)Show that g is $2\pi$ – periodic. Interpret graphically.

3)Show that, for any real , $-2\leq\,\,g(x)\leq\,\,2$ .

Exercise 3:

1)From $cos(\frac{\pi}{3})$ , determine $cos(-\frac{\pi}{3})$ then $cos(\frac{2\pi}{3})$ .

2)Same question with $sin(-\frac{\pi}{3})$ then $sin(\frac{2\pi}{3})$ .

Exercise 4:

1)Solve on $%5B0;2\pi%5B$ the equation $cos(x)=\frac{\sqrt{3}}{2}$ .

2)Solve on $%5B0;2\pi%5B$ , the equation $sin(x)=\frac{\sqrt{2}}{2}$ .

Exercise 5:

Give the abscissae of points A and B.

2)Solve on $%5B0;2\pi%5B$ , the equation $cos(x)=\frac{\sqrt{3}}{2}$ .

3)Solve on $%5B0;2\pi%5B$ , the inequation $cos(x)\,\leq\,\,\frac{\sqrt{3}}{2}$ .

Exercise 6:

In each case, check that the function f is T-periodic.

$a)f:x\,\mapsto \,cos(2\pi\,x)$ and T = 1.

$b)f:x\,\mapsto \,sin(3x)$ and $T=\frac{2\pi}{3}$ .

$c)f:x\,\mapsto \,\frac{2}{3}cos(7x+\frac{\pi}{4})$ and $T=\frac{2\pi}{7}$ .

$d)f:x\,\mapsto \,\frac{10}{7}sin(\frac{5x-8}{3})$ and $T=\frac{6\pi}{5}$ .

Exercise 7:

1.a)Determine a real x belonging to the interval $-\pi;\pi%5B$ associated to $\frac{91\pi}{4}$ .

b) Deduce $cos(\frac{91\pi}{4})$ and then, $sin(\frac{91\pi}{4})$ .

2.a)Calculate $cos(-\frac{13\pi}{6})$ .

b) Calculate $sin(-\frac{81\pi}{2})$ .

3)a)Calculate $cos(\frac{25\pi}{3})$ and derive $sin(\frac{25\pi}{3})$ .

b) Calculate $sin(\frac{45\pi}{6})$ and deduce $cos(\frac{45\pi}{6})$ .

Exercise 8:

Let f be the function defined on $%5D-\pi\,;\,\pi\,%5D$ by :

$f(x)\,=\,4cos^2(x)\,+\,2(\sqrt{2}\,-\,l)cos(x)\,-\sqrt{2}.$
The goal of the exercise is to find the solutions of the equation
f(x) = 0 and the inequation f(x) > 0.
1. Let X = cos(x).
a) Show that -1 <X< 1.
b) Show that solving the equation f(x) = 0 is
solve the equation $4X^2\,+\,2(\sqrt{2}\,-\,l)X\,-\sqrt{2}=0.$

c) Solve on [- 1 ; 1], the equation $4X^2\,+\,2(\sqrt{2}\,-\,l)X\,-\sqrt{2}=0.$
We will note and the obtained solutions.
d) Deduce the solutions on $%5D-\pi\,;\,\pi\,%5D$ of the equation f(x) = 0.
2. Let X = cos(x).
a) Solve on [-1 ; 1] the inequation $4X^2\,+\,2(\sqrt{2}\,-\,l)X\,-\sqrt{2}>0.$

Exercise 9:

1. A LP record turning 33 revolutions and $\frac{1}{3}$ of revolutions per minute contains 6 songs for a duration
total of 60 min. The duration of each song is the same.
Since the Sapphire at the end of the tone arm is located in N at the beginning of the 1st song, on which half axis will it be located at the end of the song?

2. A LP record spins 16 times and $\frac{2}{3}$ of revolutions per minute.

The duration of each song is 5 min.
The sapphire located at the end of the tone arm being located in P at the beginning of the 1st song, on which half-axis will it be located :
a) after 3 minutes?
b) after 4 minutes?
c) at the end of the 1st song?
d) at the end of the 2nd song?

Exercise 10:

Let f be the function defined on $\mathbb{R}$ by $f(x)=\frac{cos(x)}{3+sin^2(x)}$ .
1. Show that f is even and $2\pi$ -periodic.

Interpret graphically.
2. Deduce the smallest possible interval I to study f.
3. We admit that f is derivable with derivative :

$f'(x)=\frac{sin(x)(sin^2(x)-5)}{(3+sin^2(x))^2}$ .
a) Deduce the variations of the function f on l.
b) Specify the local extrema of f on l.
c) Draw the representative curve of f on [- $\pi$ ; 3 $\pi$ ].

Exercise 12:

Let f be the function defined on $\mathbb{R}$ by $f\,(x)\,=-\frac{cos^3(x)}{3}$ .
1. Show that f is even and $2\pi$ -periodic. Interpret graphically.
2. We admit that the derivative of the function f is the function defined by :
$f'(x)\,=\,cos^2(x)sin(x)$ .

a) Study the sign of .
b) Determine the direction of variation of the function f on the interval [0 ; $2\pi$ [.
c) Draw the table of variations of the function f on the interval $%5B-\pi;3\pi%5B$ .

Exercise 13:

Let us note (E) the equation $cos(x)\,=\,-x$ .
Show that the solutions of this equation belong to the interval [-1 ; 1].
2. Let f be the function defined on the interval [-1 ; 1] by f(x) = cos(x) + x.
a) Plot f with the calculator and then conjecture the number of solutions of equation (E).
Justify the approach.
b) We admit that the derivative of the function $x\,\mapsto \,cos(x)$ is the function $x\,\mapsto \,-sin(x)$ .

Deduce that $f'(x)\,=\,-sin(x)\,+\,1$ .
c) Study the sign of and deduce the direction of variation of the function f on the interval [-1 ; 1].
d) Using the calculator, give a value approximating to the nearest 0.01 of the solution(s).

Exercise 14:

The lenses located at the top of this lighthouse have a luminous range of 45 km and
a rotation time of 5 seconds.
Determine the angle covered by a lens in 1 second.
2. Calculate the area swept by a lens in 1 second.

Exercise 15:

Let m be a real non-zero parameter and the function defined on $\mathbb{R}$ by $f_m(x)\,=\,cos(mx)$ .
1. Show that is even. Interpret graphically.
2. Show that is periodic with period $T=\frac{2\pi}{m}$ .
3. Deduce that we can study on the interval $%5B0;\frac{\pi}{m}%5D$ .
4. We admit that is derivable with derivative :

$f'_m\,(x)=-msin(mx)$ . According to m :
a) Determine the sign of on the interval $%5B0;\frac{\pi}{m}%5D$ .

b) Deduce the variations of on the interval $%5B0;\frac{\pi}{m}%5D$ .
c) Draw the table of variations of on the interval $%5B0;\frac{\pi}{m}%5D$ and then on the interval $%5B-\frac{\pi}{m};\frac{2\pi}{m}%5D$ .

Exercise 16:

We consider the wind rose below.
We admit that a real having for image the direction “E” is 0 and that a real having the direction “N” is $\frac{\pi}{2}$ .

Determine a real that has as its image the direction “O”.
Determine a real with the direction “S” as its image.
Determine a real with the direction “NE” as its image.
4.a) Determine a real whose image is in the direction “NNE”.
b) By symmetry, which real can have as image the direction “SSE”?
c) By symmetry, which real can have the direction “NNW” as its image?

Exercise 17:

Calculate:

$A=cos(\frac{\pi}{4})+cos(\frac{3\pi}{4})$

$B=sin(\frac{\pi}{3})-cos(\frac{\pi}{4})+sin(\frac{5\pi}{3})$

$C=cos^2(\frac{\pi}{2})-sin^2(\frac{\pi}{2})$

Exercise 18:

Calculate:

$D=-cos(\frac{7\pi}{3})+cos(\frac{41\pi}{3})$

$B=sin(-\frac{87\pi}{4})\,+sin(-\frac{21\pi}{6})$

Exercise 19:

Exercise 20:

Let f be the function defined on $\mathbb{R}$ by $f(x)\,=\,acos(x)\,+\,bsin(x)$ .

The representative curve of f passes through the points $M(-\frac{\pi}{2};2)$ and $N(\frac{\pi}{4};\sqrt{2})$ .

1.Using the points M and N, determine the real numbers a and b.
2.deduce the expression of f as a function of x.
3. Show that f is $2\pi$ -periodic. Interpret graphically.

4. Is f even ? odd? Justify.

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Trigonometry: corrected exercises in 1st grade to download in PDF .

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