37. 1. In Δ A B C cap delta eh b c below, how does sin A compare to cos B? Is this true for the acute angles of other right triangles?

       

    2. Reading Math The word cosine is derived from the words complement's sine. Which angle in Δ A B C cap delta eh b c is the complement of ∠ A ? angle eh question mark Of ∠ B ? angle b question mark
    3. Explain why the derivation of the word cosine makes sense.

38. Proof For right Δ A B C cap delta eh b c with right ∠ C , angle c comma prove each of the following.

    1. sin A < 1 sine , eh less than , 1
    2. cos A < 1 cosine , eh less than , 1
39. 1. Writing Explain why tan 60 ° = 3 . 60 degrees equals square root of 3 . Include a diagram with your explanation.
    2. Make a Conjecture How are the sine and cosine of a 60 ° 60 degrees angle related? Explain.

The sine, cosine, and tangent ratios each have a reciprocal ratio. The reciprocal ratios are cosecant (csc), secant (sec), and cotangent (cot). Use Δ A B C cap delta eh b c and the definitions below to write each ratio.

csc X = 1 sin X sec X = 1 cos X cot X = 1 tan X co-secant x equals . fraction 1 , over sine x end fraction secant x equals . fraction 1 , over cosine x end fraction co-tangent x equals . fraction 1 , over tangent x end fraction

40. csc A co-secant eh
41. sec A secant eh
42. cot A co-tangent eh
43. csc B co-secant b
44. sec B secant b
45. cot B co-tangent b

46. Graphing Calculator Use the table begin box , table , end box feature of your graphing calculator to study sin X as X gets close to (but not equal to) 90. In the y = begin box , y equals , end box screen, enter Y 1 = sin X . cap y , 1 equals sine cap x .

    1. Use the tblset begin box , tblset , end box feature so that X starts at 80 and changes by 1. Access the table . begin box , table , end box , . From the table, what is sin X for X = 89 ? cap x equals 89 question mark
    2. Perform a “numerical zoom-in.” Use the tblset begin box , tblset , end box feature, so that X starts with 89 and changes by 0.1. What is sin X for X = 89 . 9 ? cap x equals 89 . 9 question mark
    3. Continue to zoom-in numerically on values close to 90. What is the greatest value you can get for sin X on your calculator? How close is X to 90? Does your result contradict what you are asked to prove in Exercise 38a?
    4. Use right triangles to explain the behavior of sin X found above.
47. 1. Reasoning Does tan A + tan B = tan ( A + B ) tangent , eh plus tangent , b equals tangent , open eh plus b close when A + B < 90 ? eh plus , b less than , 90 question mark Explain.
    2. Does tan A − tan B = tan ( A − B ) tangent eh minus tangent , b equals tangent open eh minus b close when A − B > 0 ? eh minus , b greater than , 0 question mark Use part (a) and indirect reasoning to explain.

C Challenge

Verify that each equation is an identity by showing that each expression on the left simplifies to 1.

48. ( sin A ) 2 + ( cos A ) 2 = 1 open sine . eh , close squared , plus open cosine . eh , close squared , equals 1
49. ( sin B ) 2 + ( cos B ) 2 = 1 open sine . b , close squared , plus open cosine . b , close squared , equals 1
50. 1 ( cos A ) 2 − ( tan A ) 2 = 1 fraction 1 , over open , cosine eh , close squared end fraction . minus . open , tangent eh , close squared . equals 1
51. 1 ( sin A ) 2 − 1 ( tan A ) 2 = 1 fraction 1 , over open , sine eh , close squared end fraction . minus . fraction 1 , over open , tangent eh , close squared end fraction . equals 1
52. Show that ( tan A ) 2 − ( sin A ) 2 = ( tan A ) 2 · ( sin A ) 2 open tangent . eh , close squared , minus open sine . eh , close squared , equals open tangent . eh , close squared , middle dot open sine . eh , close squared is an identity.

Table of Contents