Giten: \( \tan \alpha=\frac{3}{4} \); where \( \alpha \in\left[0^{\circ} ; 90^{\circ}\right] \) With the use of a sketch and without the use of a calculator, calculate. \( \sin \alpha \) \( 9.2 \cos ^{2}\left(90^{\circ}-\alpha\right)-1 \) \( 1-\sin 2 \alpha \)

To solve the problem, we start with the given condition: \[ \tan \alpha = \frac{3}{4} \] This means that in a right triangle where angle \( \alpha \) is one of the angles, the opposite side to \( \alpha \) is 3 units and the adjacent side is 4 units. We can use this information to find \( \sin \alpha \), \( \cos \alpha \), and subsequently solve the other expressions. ### Step 1: Calculate \( \sin \alpha \) Using the definition of sine in a right triangle: \[ \sin \alpha = \frac{\text{opposite}}{\text{hypotenuse}} \] First, we need to find the hypotenuse \( h \) using the Pythagorean theorem: \[ h = \sqrt{(\text{opposite})^2 + (\text{adjacent})^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] Now we can find \( \sin \alpha \): \[ \sin \alpha = \frac{3}{5} \] ### Step 2: Calculate \( 9.2 \cos^2(90^{\circ} - \alpha) - 1 \) Using the co-function identity, we know that: \[ \cos(90^{\circ} - \alpha) = \sin \alpha \] Thus, \[ \cos^2(90^{\circ} - \alpha) = \sin^2 \alpha = \left(\frac{3}{5}\right)^2 = \frac{9}{25} \] Now we can substitute this into the expression: \[ 9.2 \cos^2(90^{\circ} - \alpha) - 1 = 9.2 \cdot \frac{9}{25} - 1 \] Calculating \( 9.2 \cdot \frac{9}{25} \): \[ 9.2 \cdot \frac{9}{25} = \frac{9.2 \cdot 9}{25} = \frac{82.8}{25} = 3.312 \] Now, subtract 1: \[ 3.312 - 1 = 2.312 \] ### Step 3: Calculate \( 1 - \sin 2\alpha \) Using the double angle formula for sine: \[ \sin 2\alpha = 2 \sin \alpha \cos \alpha \] We already have \( \sin \alpha = \frac{3}{5} \). Now we need to find \( \cos \alpha \): \[ \cos \alpha = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} \] Now we can calculate \( \sin 2\alpha \): \[ \sin 2\alpha = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = 2 \cdot \frac{12}{25} = \frac{24}{25} \] Finally, we can find \( 1 - \sin 2\alpha \): \[ 1 - \sin 2\alpha = 1 - \frac{24}{25} = \frac{1}{25} \] ### Summary of Results 1. \( \sin \alpha = \frac{3}{5} \) 2. \( 9.2 \cos^2(90^{\circ} - \alpha) - 1 = 2.312 \) 3. \( 1 - \sin 2\alpha = \frac{1}{25} \) These are the final answers for the given expressions.