Download F.8 - DPS ARE

Geometry: Similarity, Right Triangles, and
Trigonometry — G-SRT
ELG.MA.HS.G.7: Define trigonometric ratios and
solve problems involving right triangles.
 G-SRT.C.6 Understand that by similarity, side
ratios in right triangles are properties of the
angles in the triangle, leading to definitions of
trigonometric ratios for acute angles.
 G-SRT.C.7 Explain and use the relationship
between the sine and cosine of complementary
angles.
 G-SRT.C.8 Use trigonometric ratios and the
Pythagorean Theorem to solve right triangles in
applied problems.★
Trigonometric Functions—F-TF
ELG.MA.HS.F.8: Extend the domain of trigonometric
functions using the unit circle.
 F-TF.A.3 (+) Use special triangles to determine
geometrically the values of sine, cosine, tangent
for π/3, π/4 and π/6, and use the unit circle to
express the values of sine, cosine, and tangent
for π–x, π+x, and 2π–x in terms of their values for
x, where x is any real number.
 F-TF.A.4 (+) Use the unit circle to explain
symmetry (odd and even) and periodicity of
trigonometric functions.
Algebra 2: Trigonometric Functions—F-TF
ELG.MA.HS.F.8: Extend the domain of
trigonometric functions using the unit circle.
 F-TF.A.1 Understand radian measure of an
angle as the length of the arc on the unit circle
subtended by the angle.
 F-TF.A.2 Explain how the unit circle in the
coordinate plane enables the extension of
trigonometric functions to all real numbers,
interpreted as radian measures of angles
traversed counterclockwise around the unit
circle.
Students will demonstrate command of the ELG by:
 Using special triangles (45°-45°-90° and 30°-60°-90°) to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6
 Using the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.
 Using the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Vocabulary:
 periodicity
 standard position
 symmetry
 trigonometric functions
 unit circle
 odd function
 even function
Sample Assessment Questions:
1) Standard(s): F-TF.A.3
Source: https://www.illustrativemathematics.org/content-standards/HSF/TF/A/3/tasks/1898
Item Prompt:
𝜋
Use the unit circle and indicated triangle below to find the exact value of the sine and cosine of the special angle .
6
Correct Answer(s):
𝜋
𝜋
First we label the vertices of the triangle that lie on the unit circle as points P and Q. We know that P has coordinates (𝑐𝑜𝑠 , 𝑠𝑖𝑛 )
6
6
Note that the blue triangle is isosceles since two of its sides are radii of the unit circle and therefore both have length 1. Therefore, the angles opposite these sides have the
𝜋
same measure, and therefore must both measure in order for the angles in the blue triangle to add up to π. This means that the blue triangle is equilateral, and so the
3
measure of segment PQ=1. But we can also observe that the two smaller triangles that make up the blue triangle are congruent by SAS (or since Q is the reflection of P across
1
1
3
√3
2
2
4
2
the x-axis); therefore, 𝑦 = , and by using the Pythagorean Theorem, we have 𝑥 2 + ( )2 = 12 , which implies 𝑥 2 = and 𝑥 =
𝜋
√3
6
2
𝑐𝑜𝑠 =
𝜋 1
and 𝑠𝑖𝑛 = .
6 2
2) Standard(s): F-TF.A.4
Source: Key Data Systems
Item Prompt:
The equation of a sine function is shown below.
y = a sin b(x – h) + k
Part A:
Write the equation of the sine function with the following attributes.
•Period: 2π
•Minimum: 2
•Maximum: 10
𝜋
•Phase shift: units to the right
2
Part B:
Is the function described in Part A even, odd, or neither?
√3 1
. Thus, P=( , ), and we conclude that
2
2
Correct Answer(s):
Part A:
𝑦 = 4𝑠𝑖𝑛 (𝑥 –
𝜋
2
) + 6
Part B:
The function is even because it is symmetric about the y–axis.