Trigonometry Overview
Objective:
1. Introduce students to trigonometric ratios using right triangles;
2. discover and analyze these ratios and solve application problems using them.
3. explore the reciprocals of the basic trigonometric ratios sine, cosine, and
tangent, along with their inverses to determine unknown angle measures.
4. Deriving the Law of Sines and the Law of Cosines to understand how to apply
trigonometry to all triangles.
1. Similar right triangles are formed by dropping vertical line segments from the
hypotenuse to the base of right triangles.
2. Given the same reference angle, the ratios opposite/hypotenuse, adjacent/
hypotenuse, and opposite/adjacent are constant.
3. The ratios opposite/hypotenuse, adjacent/hypotenuse, and opposite/adjacent
are the same for all 45-45-90 triangles and the same for all 30-60-90 triangles
given the same reference angle.
4. The slope of the hypotenuse of a 45-45-90 triangle and a 30-60-90 triangle is
equal to the opposite-to-adjacent ratio.
Reference angle- the angle of the triangle you are looking at
Opposite side- side opposite the reference angle
Adjacent angle- side adjacent to the reference angle that is not the hypotenuse
5. The tangent (tan) of an acute angle in a right triangle is the ratio of the length
of the side that is opposite the angle to the length of the side that is adjacent
to the angle. The expression “tan A” means “the tangent of <A.”
6. The cotangent (cot) of an acute angle in a right triangle is the ratio of the
length of the side that is adjacent to the angle to the length of the side that is
opposite the angle. “the expression “cot A” means the “cotangent of <A.”
7. The inverse tangent (or arc tangent) of x is the measure of an acute angle
whose tangent is x. If you know the length of any two sides of a right triangle it
is possible to compute the measure of either acute angle by using the inverse
tangent of the tan-1 button on a graphing calculator
8. The sine (sin) of an acute angle in a right triangle is the ratio of the length of
the side that is opposite the angle to the length of the hypotenuse. The
expression “sin A” means “the sine of <A”
9. The cosecant (csc) of an acute angle in a right triangle is the ratio of the
length of the hypotenuse to the length of the side that is opposite the angle.
The expression “csc “” means the “cosecant of <A”
10. The inverse sine (or arc sine) of x is the measure of an acute angle whose
sine is x. If you know the length of any two sides of a right triangle, it is
possible to calculate the measure of either acute angle by using the inverse
sine, or sin-1 button on a graphing calculator. In right triangle ABC, if sin A=x
then sin-1x=m<A
11. The cosine (cos) of an acute angle in a right triangle is the ratio of the length
of the side that is adjacent to the angle to the length of the hypotenuse. The
expression “cos A” means “the cosine of A.”
12. The secant (sec) of an acute angle in a right triangle is the ratio of the length
of the hypotenuse to the length of the side that is adjacent to the angle. The
expression “sec A” means “the secant of A.”
13. The inverse cosine (or arc cosine) of x is the measure of an acute angle
whose cosine is x. If you know the length of any two sides of a right triangle, it
is possible to compute the measure of either acute angle by using the inverse
cosine, or cos-1 button on a graphing calculator.
p. 705 to check, 706
14. When angle A and angle B are acute angles in a right triangle, sin of Angle A=
cos angle B and cos of angle A= sin of angle B
15.
When angle A and angle B are acute angles in a right triangle, csc of
angle A= sec of angle B and sec of angle A= csc of angle B
16. When angle A and angle B are acute angles in a right triangle, tan of angle A=
cot of angle B, and cot of angle A= tan of angle B
Chapter 9 summary page 729-736