Use the exact values of the ratios and find the value of [tex]$\tan 45^{\circ}+\sin 30^{\circ}$[/tex].
Answer (1)
Determine the exact value of tan 4 5 ∘ , which is 1.
Determine the exact value of sin 3 0 ∘ , which is 2 1 .
Add the two values: 1 + 2 1 = 1.5 .
The final answer is 1.5 .
Explanation
Problem Analysis We are asked to find the value of tan 4 5 ∘ + sin 3 0 ∘ using the exact values of the trigonometric ratios.
Finding tan 45 degrees The exact value of tan 4 5 ∘ is 1.
Finding sin 30 degrees The exact value of sin 3 0 ∘ is 2 1 or 0.5.
Adding the values Now, we add these two values together: tan 4 5 ∘ + sin 3 0 ∘ = 1 + 2 1 = 1 + 0.5 = 1.5
Final Answer Therefore, the value of tan 4 5 ∘ + sin 3 0 ∘ is 1.5.
Examples
Imagine you are building a ramp for a skateboard park. The angle of the ramp is 30 degrees, and you need to calculate the horizontal distance covered for every unit of vertical height. This involves understanding the sine of the angle. Also, you are designing a square garden and want to know the ratio of the side length to the distance from one corner to the opposite corner, which involves understanding the tangent of 45 degrees. Combining these trigonometric concepts helps in designing structures and understanding spatial relationships.
