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Here are a few recommended readings before getting started with this lesson.
Tiffaniqua, who works as a landscape designer, received a job to create a new design for an old city park. Since the park is quite huge, she divided its area into six rectangular sections. The first section contains a fountain (F) and is crossed by a river at two points — south (S) and north (N).
When she first came to analyze the park, she stood at the north-west corner of the first section, which she marked as point M. Then, she took notes of some measures of angles and distances.
Later when returning to her work space, Tiffaniqua used her notes to make additional calculations. What is the length of the river within the first section of the park? Round the answer to the first decimal place.
There are identities that allow calculating the values of trigonometric functions of the sum or difference of two angles.
To evaluate trigonometric functions of the sum of two angles, the following identities can be applied.
There are also similar identities for the difference of two angles.
Let △AFD be a right triangle with hypotenuse 1 and an acute angle with measure x+y.
By definition, the sine of an angle is the ratio between the lengths of the opposite side and the hypotenuse.By the Third Angle Theorem, it is known that ∠GAF≅∠GDC. Therefore, m∠GDC=y.
Since the purpose is to rewrite DF, plot a point E on DF such that EC∥AB. This way a rectangle ECBF is formed. The opposite sides of a rectangle have the same length, so EF and CB are equal. Also, CE⊥DF makes △CED a right triangle.
Consequently, EF=cosxsiny and DE can be written in terms of sinx and cosy using the cosine ratio.Consider the following process for calculating the exact value of sin120∘.
Rewrite 120∘ as 90∘+30∘
sin(x+y)=sinxcosy+cosxsiny
Substitute values
1⋅a=a
Zero Property of Multiplication
Identity Property of Addition
When Tiffaniqua came home from work, she saw that her son Davontay and his friend Zain came up with a game. Davontay assigned numbers 1 through 6 to the trigonometric functions of sine, cosine, and tangent, while Zain assigned numbers 1 through 6 to six angle measures.
Next, they rolled the dice four times to identify which two trigonometric values each person should calculate. The die on the left determines the trigonometric function and the die on the right determines the angle measure.x=60∘, y=45∘
Add terms
Multiply fractions
Add fractions
x=240∘, y=45∘
Subtract term
tan(240∘)=3
a⋅1=a
ba=b⋅(1−3)a⋅(1−3)
(a+b)(a−b)=a2−b2
Multiply parentheses
Calculate power
Add and subtract terms
Put minus sign in front of fraction
Simplify quotient
Distribute -1
Commutative Property of Addition
x=120∘, y=45∘
Subtract term
Multiply fractions
Commutative Property of Addition
Subtract fractions
x=30∘, y=45∘
Subtract term
a⋅1=a
Rewrite 1 as 33
Add fractions
ba/dc=ba⋅cd
Multiply fractions
ba=b/3a/3
ba=b⋅(3−3)a⋅(3−3)
(a+b)(a−b)=a2−b2
a⋅a=a2
(a−b)2=a2−2ab+b2
Calculate power
Add and subtract terms