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If cosx=siny, then which of the following pairs of angle measures could NOT be the values of x and y, respectively? a.π4,π4 b.π6,π3 c.π8,3π8 d.π2,π2

Short Answer

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

Step by step solution

01

- Recall trigonometric identities

Recall that cosx=siny is true when y=π/2x. This relationship shows that the sine of an angle is equal to the cosine of its complementary angle.
02

- Verify pair (a)

Check if y=π4 when x=π4. \ \Here, sinπ4=cosπ4 which is true because both equal 22 . Thus, π4,π4 is a valid pair.
03

- Verify pair (b)

Check if y=π3 when x=π6. \ \We see that sinπ3=cosπ6, which is true because both equal 32 . Thus, π6,π3 is also a valid pair.
04

- Verify pair (c)

Check if y=3π8 when x=π8. \ \We need to ensure sin3π8=cosπ8. Calculating, cosπ8 does not equal sin3π8. Thus, π8,3π8 is NOT a valid pair.
05

- Verify pair (d)

Check if y=π2 when x=π2. \ \Here, cosπ2 is 0 and sinπ2 is 1, which means cosx\eqsiny in this case. Thus, π2,π2 is NOT a valid pair.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

complementary angles
Complementary angles are two angles that add up to 90 degrees or π2 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=π2x, then siny=cosx. 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 π/2x.
  • 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 θ in a right triangle:
  • sinθ is the ratio of the length of the opposite side to the hypotenuse.
  • cosθ is the ratio of the length of the adjacent side to the hypotenuse.
  • tanθ is the ratio of the length of the opposite side to the adjacent side.

Examples:
  • sin(π4)=cos(π4)=22
  • sin(π6)=cos(π3)=12
  • sin(π3)=cos(π6)=32
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 2π radians. Therefore, one degree equals π180 radians, and one radian equals 180π degrees.

Converting between degrees and radians is straightforward:
  • To convert degrees to radians, multiply the degree measure by π180.
  • To convert radians to degrees, multiply the radian measure by 180π.

For example:
  • 30=30×π180=π6radians
  • π4radians=π4×180π=45
Understanding these conversions and measures assists in solving trigonometric identities and other mathematical problems.

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