21. If [tex]\cos q = x[/tex] and [tex]\sin 60^{\circ}= x +0.5,0^{\circ}\ \textless \ q \ \textless \ 90^{\circ}[/tex] find correct to the nearest degree the value of q.
(a) [tex]66^{\circ}[/tex]
(b) [tex]67^0[/tex]
(c) [tex]68^0[/tex]
(d) [tex]69^{\circ}[/tex]
The table shows the ages of students in a club. Use it to answer questions 22 and 23.
Answer (1)
Find sin 6 0 ∘ which is approximately 0.8660 .
Solve for x using the equation sin 6 0 ∘ = x + 0.5 , resulting in x = 0.3660 .
Substitute x into cos q = x , so cos q = 0.3660 .
Find q by taking the inverse cosine of 0.3660 , which gives q ≈ 68.5 3 ∘ , and round to the nearest degree, resulting in 6 9 ∘ .
Explanation
Understanding the Problem We are given that cos q = x and sin 6 0 ∘ = x + 0.5 , where 0 ∘ < q < 9 0 ∘ . We need to find the value of q to the nearest degree.
Finding sin 60 degrees First, we find the value of sin 6 0 ∘ . We know that sin 6 0 ∘ = 2 3 ≈ 0.8660 .
Solving for x Next, we substitute the value of sin 6 0 ∘ into the equation sin 6 0 ∘ = x + 0.5 and solve for x . So, 0.8660 = x + 0.5 . Subtracting 0.5 from both sides, we get x = 0.8660 − 0.5 = 0.3660 .
Substituting x into cos q = x Now, we substitute the value of x into the equation cos q = x . So, cos q = 0.3660 .
Finding q To find the value of q , we take the inverse cosine (arccos) of x . So, q = arccos ( 0.3660 ) . Using a calculator, we find that q ≈ 68.5 3 ∘ .
Rounding to Nearest Degree Finally, we round the value of q to the nearest degree. Since 68.5 3 ∘ is closer to 6 9 ∘ than 6 8 ∘ , we round up to 6 9 ∘ .
Examples
Imagine you're building a ramp and need to determine the angle at which the ramp rises. Knowing the relationship between angles and trigonometric functions like cosine helps you calculate the correct angle for the ramp, ensuring it meets safety standards. This problem demonstrates how trigonometric functions are used in real-world applications to calculate angles and distances.
