1.) $$ \sin 60^{\circ} \cdot \cos 45^{\circ} $$ 2.) $$ \sin 30^{\circ}+\cos 60^{\circ} $$ 3.) $$ 4 \sin 30^{\circ}+8 an 45^{\circ} $$ 4.) $$ \cos ^{2} 45+\sin ^{2} 45^{\circ} $$ 5.) $$ \sin ^{2} 60+\cos ^{2} 45^{\circ}+\sin ^{2} 30^{\circ} $$ (.) $$ 3{ an 45^{\circ}+2 \sin 30^{\circ}}_{ ext {7.) } \sin ^{2} 60^{\circ} imes 2 an 49^{\circ}}^{ ext {8.) } 2 \cos 90^{\circ}+3 \sin 90^{\circ}} $$ 9.) $$ \frac{1}{3} an 60^{\circ}+3 an 30^{\circ} $$ 10.) $$ 2 \sin 45^{\circ} \cdot \cos 45^{\circ} $$

Question

1.) $$ \sin 60^{\circ} \cdot \cos 45^{\circ} $$ 2.) $$ \sin 30^{\circ}+\cos 60^{\circ} $$ 3.) $$ 4 \sin 30^{\circ}+8 	an 45^{\circ} $$ 4.) $$ \cos ^{2} 45+\sin ^{2} 45^{\circ} $$ 5.) $$ \sin ^{2} 60+\cos ^{2} 45^{\circ}+\sin ^{2} 30^{\circ} $$ (.) $$ 3{	an 45^{\circ}+2 \sin 30^{\circ}}_{	ext {7.) } \sin ^{2} 60^{\circ} 	imes 2 	an 49^{\circ}}^{	ext {8.) } 2 \cos 90^{\circ}+3 \sin 90^{\circ}} $$ 9.) $$ \frac{1}{3} 	an 60^{\circ}+3 	an 30^{\circ} $$ 10.) $$ 2 \sin 45^{\circ} \cdot \cos 45^{\circ} $$

UpStudy AI · Accepted Answer

Alright, I need to solve these trigonometric expressions. Let's tackle them one by one, ensuring I use the correct values for each trigonometric function at the given angles.

1. **$$ \sin 60^{\circ} \cdot \cos 45^{\circ} $$**
   - I recall that $$ \sin 60^{\circ} = \frac{\sqrt{3}}{2} $$ and $$ \cos 45^{\circ} = \frac{\sqrt{2}}{2} $$.
   - Multiplying these together: $$ \frac{\sqrt{3}}{2} 	imes \frac{\sqrt{2}}{2} = \frac{\sqrt{6}}{4} $$.

2. **$$ \sin 30^{\circ} + \cos 60^{\circ} $$**
   - $$ \sin 30^{\circ} = \frac{1}{2} $$ and $$ \cos 60^{\circ} = \frac{1}{2} $$.
   - Adding them: $$ \frac{1}{2} + \frac{1}{2} = 1 $$.

3. **$$ 4 \sin 30^{\circ} + 8 	an 45^{\circ} $$**
   - $$ \sin 30^{\circ} = \frac{1}{2} $$ and $$ 	an 45^{\circ} = 1 $$.
   - Calculating: $$ 4 	imes \frac{1}{2} + 8 	imes 1 = 2 + 8 = 10 $$.

4. **$$ \cos^{2} 45^{\circ} + \sin^{2} 45^{\circ} $$**
   - Using the Pythagorean identity: $$ \cos^{2} 	heta + \sin^{2} 	heta = 1 $$.
   - Therefore, $$ \cos^{2} 45^{\circ} + \sin^{2} 45^{\circ} = 1 $$.

5. **$$ \sin^{2} 60^{\circ} + \cos^{2} 45^{\circ} + \sin^{2} 30^{\circ} $$**
   - Calculating each term:
     - $$ \sin^{2} 60^{\circ} = \left( \frac{\sqrt{3}}{2} ight)^2 = \frac{3}{4} $$
     - $$ \cos^{2} 45^{\circ} = \left( \frac{\sqrt{2}}{2} ight)^2 = \frac{2}{4} = \frac{1}{2} $$
     - $$ \sin^{2} 30^{\circ} = \left( \frac{1}{2} ight)^2 = \frac{1}{4} $$
   - Adding them up: $$ \frac{3}{4} + \frac{1}{2} + \frac{1}{4} = \frac{3}{4} + \frac{2}{4} + \frac{1}{4} = \frac{6}{4} = 1.5 $$.

6. **$$ 3 	an 45^{\circ} + 2 \sin 30^{\circ} $$**
   - $$ 	an 45^{\circ} = 1 $$ and $$ \sin 30^{\circ} = \frac{1}{2} $$.
   - Calculating: $$ 3 	imes 1 + 2 	imes \frac{1}{2} = 3 + 1 = 4 $$.

7. **$$ \sin^{2} 60^{\circ} 	imes 2 	an 49^{\circ} $$**
   - $$ \sin^{2} 60^{\circ} = \frac{3}{4} $$.
   - $$ 	an 49^{\circ} $$ is approximately $$ 1.1504 $$.
   - Calculating: $$ \frac{3}{4} 	imes 2 	imes 1.1504 \approx 1.7256 $$.

8. **$$ 2 \cos 90^{\circ} + 3 \sin 90^{\circ} $$**
   - $$ \cos 90^{\circ} = 0 $$ and $$ \sin 90^{\circ} = 1 $$.
   - Calculating: $$ 2 	imes 0 + 3 	imes 1 = 0 + 3 = 3 $$.

9. **$$ \frac{1}{3} 	an 60^{\circ} + 3 	an 30^{\circ} $$**
   - $$ 	an 60^{\circ} = \sqrt{3} $$ and $$ 	an 30^{\circ} = \frac{\sqrt{3}}{3} $$.
   - Calculating: $$ \frac{1}{3} 	imes \sqrt{3} + 3 	imes \frac{\sqrt{3}}{3} = \frac{\sqrt{3}}{3} + \sqrt{3} = \frac{\sqrt{3}}{3} + \frac{3\sqrt{3}}{3} = \frac{4\sqrt{3}}{3} $$.

10. **$$ 2 \sin 45^{\circ} \cdot \cos 45^{\circ} $$**
    - $$ \sin 45^{\circ} = \frac{\sqrt{2}}{2} $$ and $$ \cos 45^{\circ} = \frac{\sqrt{2}}{2} $$.
    - Multiplying: $$ 2 	imes \frac{\sqrt{2}}{2} 	imes \frac{\sqrt{2}}{2} = 2 	imes \frac{2}{4} = 1 $$.

1. $$ \frac{\sqrt{6}}{4} $$ 2. $$ 1 $$ 3. $$ 10 $$ 4. $$ 1 $$ 5. $$ 1.5 $$ 6. $$ 4 $$ 7. $$ \approx 1.7256 $$ 8. $$ 3 $$ 9. $$ \frac{4\sqrt{3}}{3} $$ 10. $$ 1 $$

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Trigonometry Questions