Name: ____________________ Pre- Calc. 11 Date: _______________

Chapter 2– Trigonometry Section 2.1- Angles in Standard Position

Key Ideas: The three angles in a triangle sum to ___________.

Pythagorean Theorem: __________________________ where c is always the __________________.

The three trigonometric ratios for right triangles are:

Focus On: Sketching an angle from 00 - 3600 in standard position and determining its reference

angle. Determining the quadrant in which an angle in standard position terminates.

Determining the exact values of the sine, cosine, and tangent ratios of a given angle with reference angle 30°, 45°, 60°.

Solving problems involving trigonometric ratios.

An angle that is drawn in standard position must have its vertex at the origin of the Cartesian plane, and its initial arm must coincide with the positive 𝑥-axis.

Angle in Standard Position Label the four quadrants of a Cartesian plane:

Example 1: Draw each angle in standard position and identify the quadrant in which it lies:

a) 60 b) 100 c) 300

SINE COSINE TANGENT

sin 𝜃 =𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 cos 𝜃 =

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 tan 𝜃 =

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

S O H C A H T O A

For each angle in standard position, there is a corresponding acute angle called the reference angle, which is the acute angle between the terminal arm and the (nearest) x-axis. Thus, any reference angle is between 0 and 90

Example 2: Draw each angle in standard position, and find the reference angle.

a) 30 b) 250 c) 325 d) 100

Example 3: Determine the angle in standard position when an angle of 60 is reflected

a) in the y-axis b) in the x-axis c) in the y-axis & then in the x-axis

Special Right Triangles

A 45-45-90 triangle with legs of each 1 unit has a hypotenuse of √2.

The trigonometric ratios on the previous page are given as exact values (in fraction/radical form as opposed to an approximated decimal).

A 30-60-90 triangle has legs of 1 unit and √3 units, with a hypotenuse of 2 units.

Example 4: Suppose you have a standard angle of 60 and the initial arm is 6.75 meters. What is the

length of the terminal arm?

Example 5: Allie is learning to play the piano. Her teacher uses a metronome to help her keep time.

The pendulum arm of the metronome is 10 cm long. For one particular tempo, the setting results in the arm moving back and forth from a start position of 60° to 120°. What horizontal distance does the tip of the arm move in one beat? Give an exact answer.

Assignment: Pg. 83-87 Q #1, 2, 3ace, 4bd, 5, 6a, 7abc, 8, 13

𝑠𝑖𝑛𝜃 =𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑐𝑜𝑠𝜃 =

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑡𝑎𝑛𝜃 =

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

S O H C A H T O A

For the 45-45-90 triangle,

𝑠𝑖𝑛45° =

𝑐𝑜𝑠45° =

𝑡𝑎𝑛45° =

For the 30-60-90 triangle,

𝑠𝑖𝑛30° =

𝑐𝑜𝑠30° =

𝑡𝑎𝑛30° =

𝑠𝑖𝑛60° =

𝑐𝑜𝑠60° =

𝑡𝑎𝑛60° =

Name: ____________________ Pre- Calc. 11 Date: _______________

Chapter 2– Trigonometry Review Worksheet FOR THIS PAGE, ROUND ANSWERS TO THE NEAREST TENTH WHEN APPROPTIATE.

1. Solve each triangle.

a)

b)

c)

2. Solve XYZ is Y = 90:

a) XY = 24 and XZ = 35

b) XZ = 72 and Z = 52

c) YZ = 32 and X = 64

3. Find the measure of A to the nearest degree if each value is a) sin A and b) cos A.

(i) 0.2079 (ii) 0.4384 (iii) 0.7431 (iv) 0.9063 (v) 0.9945

G

H K 55

18

P

Q

R

21

27

C

B A

30 cm 18 cm

4. Write the exact values of sin A, cos A, and tan A for:

a)

b)

5. Find the other two primary trigonometric ratios (exact values) of if:

a) sin = 17

8 b) cos = 25

7

c) tan = 21

20 d) tan = 3

2

e) cos = 8

55 f) sin =

32

3

6. Draw each angle in standard position. Determine and label its reference angle.

a) 50 b) 120 c) 165 d) 240 e) 90

f) 45 g) 60 h) 75 i) 155 j) 230

7. P is a point on the terminal arm of . Use a diagram to determine the three primary trigonometric ratio

values of and the three reciprocal trigonometric ratio values of . Find exact values.

a) P(12,5) b) P(4,2) c) P(3,1) d) P(3,4)

e) P(6,2) f) P(2,9) g) P(0,4) d) P(5,0)

A

53 28

45

A

15 8

17