Question

If $cosx=\sin y,$ then which of the following pairs of angle measures could NOT be the values of $x$ and $y,$ respectively? a.$\frac{\pi }{4},\frac{\pi }{4}$ b.$\frac{\pi }{6},\frac{\pi }{3}$ c.$\frac{\pi }{8},\frac{3\pi }{8}$ d.$\frac{\pi }{2},\frac{\pi }{2}$

Short answer

Pairs (c) and (d) are not valid pairs.

Step-by-step solution

- Recall trigonometric identities

Recall that $\cos x=\sin y$ is true when $y=\pi /2-x$. This relationship shows that the sine of an angle is equal to the cosine of its complementary angle.

- Verify pair (a)

Check if $y=\frac{\pi }{4}$ when $x=\frac{\pi }{4}$. \ \Here, $\sin \frac{\pi }{4}=\cos \frac{\pi }{4}$ which is true because both equal $\frac{\sqrt{2}}{2}$ . Thus, $\frac{\pi }{4},\frac{\pi }{4}$ is a valid pair.

- Verify pair (b)

Check if $y=\frac{\pi }{3}$ when $x=\frac{\pi }{6}$. \ \We see that $\sin \frac{\pi }{3}=\cos \frac{\pi }{6}$, which is true because both equal $\frac{\sqrt{3}}{2}$ . Thus, $\frac{\pi }{6},\frac{\pi }{3}$ is also a valid pair.

- Verify pair (c)

Check if $y=\frac{3\pi }{8}$ when $x=\frac{\pi }{8}$. \ \We need to ensure $\sin \frac{3\pi }{8}=\cos \frac{\pi }{8}$. Calculating, $\cos \frac{\pi }{8}$ does not equal $\sin \frac{3\pi }{8}$. Thus, $\frac{\pi }{8},\frac{3\pi }{8}$ is NOT a valid pair.

- Verify pair (d)

Check if $y=\frac{\pi }{2}$ when $x=\frac{\pi }{2}$. \ \Here, $\cos \frac{\pi }{2}$ is 0 and $\sin \frac{\pi }{2}$ is 1, which means $\cos x\text{eq}\sin y$ in this case. Thus, $\frac{\pi }{2},\frac{\pi }{2}$ is NOT a valid pair.

Key concepts

complementary angles

Complementary angles are two angles that add up to 90 degrees or $\frac{\pi }{2}$ radians. These angles have a special property in trigonometry: the sine of one angle is equal to the cosine of the other. For example, if we have angles $x$ and $y$ where $y=\frac{\pi }{2}-x$, then $\sin y=\cos x$. This relationship is extremely helpful in simplifying trigonometric expressions and solving equations.

Some important points about complementary angles:

• If angle $x$ is known, angle $y$ can be easily found as $\pi /2-x$.
• The functions sine and cosine exhibit a complementary relationship which makes them interchangeable under certain conditions.
• Complementary angles are often used in right triangle trigonometry since the two non-right angles in a right triangle are always complementary.

trigonometric functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). For an angle $\theta$ in a right triangle:

• $\sin \theta$ is the ratio of the length of the opposite side to the hypotenuse.
• $\cos \theta$ is the ratio of the length of the adjacent side to the hypotenuse.
• $\tan \theta$ is the ratio of the length of the opposite side to the adjacent side.

Examples:

• $\sin (\frac{\pi }{4})=\cos (\frac{\pi }{4})=\frac{\sqrt{2}}{2}$
• $\sin (\frac{\pi }{6})=\cos (\frac{\pi }{3})=\frac{1}{2}$
• $\sin (\frac{\pi }{3})=\cos (\frac{\pi }{6})=\frac{\sqrt{3}}{2}$

Understanding trigonometric functions helps solve problems involving right triangles, periodic phenomena, and oscillatory systems.

angle measures

Angle measures can be given in degrees or radians. Degrees are more common in everyday use, while radians are used frequently in mathematics and physics. A full circle is 360 degrees or $2\pi$ radians. Therefore, one degree equals $\frac{\pi }{180}$ radians, and one radian equals $\frac{180}{\pi }$ degrees.

Converting between degrees and radians is straightforward:

• To convert degrees to radians, multiply the degree measure by $\frac{\pi }{180}$.
• To convert radians to degrees, multiply the radian measure by $\frac{180}{\pi }$.

For example:

• ${30}^{\circ }=30\times \frac{\pi }{180}=\frac{\pi }{6}\mathrm{radians}$
• $\frac{\pi }{4}\mathrm{radians}=\frac{\pi }{4}\times \frac{180}{\pi }={45}^{\circ }$

Understanding these conversions and measures assists in solving trigonometric identities and other mathematical problems.