Complete the sentences.

• The sine of the acute angleacute angleacute angle in a right‑angled triangleright‑angled triangleright‑angled triangle is called ...

• The cosine of the acute angle in a right‑angled triangle is called ...

• The tangent of the acute angleacute angleacute angle in a right‑angled triangleright‑angled triangleright‑angled triangle is called ...

• The value of the sine and the cosine of the acute angle belongs to the range of ...

The aim of the lesson is to calculate the numerical values of trigonometric expressions. Analyze the material presented in the interactive drawing.

Think - Can the values of other trigonometric functionstrigonometric functionstrigonometric functions be determined when one function is given? Formulate appropriate conclusions.

Angle $\alpha$ is an acute angleacute angleacute angle, $\sin \alpha =0,4$. Calculate the values of the other trigonometric functionstrigonometric functionstrigonometric functions of this angle.

Angle $\alpha$ is an acute angle, $\cos \alpha =0,8$. Calculate the values of the other trigonometric functions of this angle.

Angle $\alpha$ is an acute angleacute angleacute angle, $tg\alpha =6$. Calculate the values of the other trigonometric functionstrigonometric functionstrigonometric functions of this angle.

Calculate the values of the expressions:

a) ${(sin{18}^{\circ })}^2+{(sin{72}^{\circ })}^2+3$

b) $sin{35}^{\circ }\cdot cos{55}^{\circ }+cos{35}^{\circ }\cdot sin{55}^{\circ }$

c) $\frac{{({(sin{60}^{\circ })}^2+{(cos{30}^{\circ })}^2)}^2-({(sin{20}^{\circ })}^2+{(sin{70}^{\circ })}^2)}{(sin{45}^{\circ }\cdot cos{45}^{\circ })}$

Check if there is an acute angleacute angleacute angle $\alpha$ that satisfies the given condition:

a) $\cos \alpha =4-\cos \alpha$

b) $6+3·sin\alpha =3$

c) $\sqrt{8}-3·sin\alpha +5=8·sin\alpha$

Knowing, that $sin\alpha +cos\alpha =1,25$, calculate $sin\alpha ·cos\alpha$.

An extra task:

Prove, that there is no acute $\alpha$ such that $\frac{\sin \alpha }{tg\alpha }+\cos \alpha =5$.

In a right‑angled triangle, the product of the sine of one acute angle and the tangent of the second acute angle is equal to $\frac{\sqrt{3}}{2}$. The hypotenuse is $6\sqrt{3}$ long.

Calculate the legs and acute angles of this triangle.

Describe in English the relations between trigonometric functions of acute angles in the same right‑angled triangle.