Question

consider the relationship between the sine and cosine of complementary angles.
a. the sine of an angle is equal to the cosine of its complement, and the cosine of an angle is equal to the sine of its complement.
b. use the relationship to find an equivalent expression for each expression shown.
i. sin 68°
ii. cos 13°
iii. sin 49°
iv. cos 65°
v. sin 27°
vi. cos 32°
c. verify with a calculator that each expression and its equivalent from part b have the same value. round the value of each expression to the nearest ten - thousandth.

Explanation:

Step1: Recall the relationship

If \(\theta\) is an angle, then \(\sin\theta=\cos(90^{\circ}-\theta)\) and \(\cos\theta = \sin(90^{\circ}-\theta)\)

Step2: Find equivalent expressions

• For \(\sin68^{\circ}\):

Since \(\sin\theta=\cos(90^{\circ}-\theta)\), when \(\theta = 68^{\circ}\), \(\sin68^{\circ}=\cos(90^{\circ}-68^{\circ})=\cos22^{\circ}\)

• For \(\cos13^{\circ}\):

Since \(\cos\theta=\sin(90^{\circ}-\theta)\), when \(\theta = 13^{\circ}\), \(\cos13^{\circ}=\sin(90^{\circ}-13^{\circ})=\sin77^{\circ}\)

• For \(\sin49^{\circ}\):

Since \(\sin\theta=\cos(90^{\circ}-\theta)\), when \(\theta = 49^{\circ}\), \(\sin49^{\circ}=\cos(90^{\circ}-49^{\circ})=\cos41^{\circ}\)

• For \(\cos65^{\circ}\):

Since \(\cos\theta=\sin(90^{\circ}-\theta)\), when \(\theta = 65^{\circ}\), \(\cos65^{\circ}=\sin(90^{\circ}-65^{\circ})=\sin25^{\circ}\)

• For \(\sin27^{\circ}\):

Since \(\sin\theta=\cos(90^{\circ}-\theta)\), when \(\theta = 27^{\circ}\), \(\sin27^{\circ}=\cos(90^{\circ}-27^{\circ})=\cos63^{\circ}\)

• For \(\cos32^{\circ}\):

Since \(\cos\theta=\sin(90^{\circ}-\theta)\), when \(\theta = 32^{\circ}\), \(\cos32^{\circ}=\sin(90^{\circ}-32^{\circ})=\sin58^{\circ}\)

Step3: Verify with calculator

• \(\sin68^{\circ}\approx0.9272\), \(\cos22^{\circ}\approx0.9272\)
• \(\cos13^{\circ}\approx0.9744\), \(\sin77^{\circ}\approx0.9744\)
• \(\sin49^{\circ}\approx0.7547\), \(\cos41^{\circ}\approx0.7547\)
• \(\cos65^{\circ}\approx0.4226\), \(\sin25^{\circ}\approx0.4226\)
• \(\sin27^{\circ}\approx0.4540\), \(\cos63^{\circ}\approx0.4540\)
• \(\cos32^{\circ}\approx0.8480\), \(\sin58^{\circ}\approx0.8480\)

Answer:

i. \(\cos22^{\circ}\)
ii. \(\sin77^{\circ}\)
iii. \(\cos41^{\circ}\)
iv. \(\sin25^{\circ}\)
v. \(\cos63^{\circ}\)
vi. \(\sin58^{\circ}\)