Trigonometry : 11th grade math worksheets with answers in PDF

11th grade math worksheets

Math worksheets in 11th grade on metric relations in any triangle.

Exercise 1 – Trigonometric equations
Solve in $%5D-\pi;\pi%5D$ the following equations.
1. $cosx=\frac{\sqrt{2}}{2}.$
2. $sinx=-\frac{\sqrt{3}}{2}.$

Exercise 2 – Determine the value of cosine
In this exercise, we give:
$cos(\frac{\pi}{5})=\frac{1+\sqrt{5}}{4}.$
Calculate the exact value of $cos(\frac{2\pi}{5})$ and then of $cos(\frac{3\pi}{5}).$

Exercise 3 – Exercise on the tangent
In this exercise, we have the following data: $tan(\frac{\pi}{12})=2-\sqrt{3}.$
1. Let $x\in%5D0;\frac{\pi}{2}%5B$ . Prove that $tan(\frac{\pi}{2}-x)=\frac{1}{tanx}.$
2. Deduce that :
$tan(\frac{5\pi}{12})=2+\sqrt{3}.$

Exercise 4 – Solving a trigonometric equation
Solve in $%5D-\pi;\pi%5D$ the equation: sin(2x) = cos(x).

Exercise 5 – Solving two trigonometric equations
Solve in $%5D-\pi;\pi%5B$ the following equations:
$1.2cos^3x-7cos^2x+2cosx+3=0.\\2.2sin^3x+cos^2x5sinx-3=0.$

Exercise 6 – Solving a complex trigonometric equation
Solve in $\mathbb{R}$ the equation :

Exercise 7 – Equilateral triangle and angle measurement
In the figure below, ABC is equilateral, BCI and ACJ are isosceles rectangles at B and J respectively.

1. Determine a measure of each of the following angles:
$\,(\,\vec{AB},\vec{AC\,}\,)$
$\,(\,\vec{BI},\vec{BA\,}\,)$
$\,(\,\vec{AI},\vec{AB\,}\,)$
$\,(\,\vec{BC},\vec{CJ}\,)$
$\,(\,\vec{CJ},\vec{BI}\,)$
2. Show that the points A, I and J are aligned.

Exercise 8 – Trigonometric circle and points
Draw a trigonometric circle and place on this circle
the points A, M, N, P and Q marked by the following numbers :
$0,\frac{2\pi}{3},\frac{7\pi}{6},\frac{-3\pi}{4},-\frac{7\pi}{3}$

Exercise 9 – Principal measure of an angle
Determine the principal measure of angles:
$\alpha\,=\frac{114\pi}{4}$ ; $\beta\,=-\frac{91\pi}{6}$ ; $\gamma\,=70$

Exercise 10 – Metric relations in the triangle
ABC is a triangle with $BC=4,\widehat{B}=\frac{\pi}{4};\widehat{C}=\frac{\pi}{3}$ .
1. Prove that $sin\,\,\,\widehat{A}=\frac{\sqrt{6}+\sqrt{2}}{4}$ .
2. Calculate the exact values of AB and AC .

Exercise 11 – Graphical representation of trigonometric functions
Show that the graphical representation of the function defined on $\mathbb{R}$ by :

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

Exercise 12 – Solving a trigonometric equation
Show that, for any real :

Exercise 13 – Using addition formulas
Using the addition formulas, calculate the exact value of $sin(\frac{7\pi}{12})\,et\,cos(\frac{7\pi}{12}).$

Exercise 14 – The Al-Kashi formulas
Let ABC be any triangle.
We note:
– has the length of the segment [BC];
– b the length of the segment [AC];
– c the length of the segment [AB];

Show that :
$a^2=b^2+c^2-2bc\times \,cos(\alpha\,)\\b^2=a^2+c^2-2ac\times \,cos(\beta\,)\\c^2=a^2+b^2-2ab\times \,cos(\gamma\,)\\$

Exercise 15 – Trigonometry formula
Show that:
$\frac{sin3x}{sinx}+\frac{cos3x}{cosx}=4cos2x$

Exercise 16 – Main measure and figure in the plane
Compute the principal measure of $(\vec{DC},\vec{DA})$ knowing that :
$(\vec{AB},\vec{AD})=\frac{\pi}{6}\,\,%5B2\pi%5D$
$(\vec{BC},\vec{BA})=\frac{\pi}{4}\,\,%5B2\pi%5D$
$(\vec{CD},\vec{CB})=\frac{\pi}{8}\,\,%5B2\pi%5D$

Exercise 17 – Metric relations in the triangle

ABC is a triangle with BC = 4, $\widehat{B}=\frac{\pi}{4}$ and $\widehat{C}=\frac{\pi}{3}$ .
1. Prove that $sin\widehat{A}=\frac{\sqrt{6}+\sqrt{2}}{4}$ .
2. Calculate the exact values of AB and AC.
3. Calculate the exact value of the area of ABC.

Exercise 18
A triangle ABC has area S = 5 cm².
Moreover, c=AB=13 cm and b=AC= 2 cm.
Calculate the possible length(s) of the third side a = BC.

Exercise 19
ABC is a triangle.
We know that AB = 7, AC= 4 and $\widehat{A}=60^{\circ}$ .
1. Calculate the exact value of BC.
2. Calculate the exact value of $sin\widehat{B}$ .

Exercise 20
Show that two supplementary angles have the same sine.
ABCD is a quadrilateral.We assume that the segments [AC] and [BD] are inside the quadrilateral.
Show that the area S of quadrilateral ABCD is given by :
$S=\frac{1}{2}\,\times \,AC\,\times \,BD\,\times \,sin\Theta$ ( $\Theta$ is the angle formed by the diagonals).

Exercise 21
A walker walks 5 km east, then 2 km northeast, surprised by the weather, he runs straight back to his starting point.
What distance did he run?
We will give the exact value and then the approximate value to the nearest 0.01 km.

Exercise 22
Prove the following property:
ABC is a right-angled triangle in A $\Leftrightarrow$ .

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