Use the fundamental identities to find the value of the trigonometric function.
-An airplane flying faster than the speed of sound creates sound waves that form a cone. If $\theta$ is the vertex angle of the cone and $\mathrm { m }$ is the Mach number for the speed of the plane, then $\sin \frac { \theta } { 2 } = \frac { 1 } { \mathrm {~m} }$ $( \mathrm { m } 1 )$. Find $\theta$ to the nearest degree if $\mathrm { m } = 2.9$.
A) $45 ^ { \circ }$
B) $43 ^ { \circ }$
C) $38 ^ { \circ }$
D) $40 ^ { \circ }$

Question

Use the fundamental identities to find the value of the trigonometric function.
-An airplane flying faster than the speed of sound creates sound waves that form a cone. If $\theta$ is the vertex angle of the cone and $\mathrm { m }$ is the Mach number for the speed of the plane, then $\sin \frac { \theta } { 2 } = \frac { 1 } { \mathrm {~m} }$ $( \mathrm { m } > 1 )$. Find $\theta$ to the nearest degree if $\mathrm { m } = 2.9$.
A) $45 ^ { \circ }$
B) $43 ^ { \circ }$
C) $38 ^ { \circ }$
D) $40 ^ { \circ }$

Quizplus · Accepted Answer

Given $\sin \frac { \theta } { 2 } = \frac { 1 } { \mathrm {~m} }$ and $\mathrm { m } = 2.9$, we find $\sin \frac { \theta } { 2 } = \frac { 1 } { 2.9 }$. Solving for $\theta$, we get $\theta = 2 \times \sin^{-1}\left(\frac{1}{2.9}\right)$ which approximates to $40^\circ$.

Use the Fundamental Identities to Find the Value of the Trigonometric