6.2 Trig. Functions of Acute Angles
Pre-Calculus
Learning Targets
1. Know and apply the six trigonometric ratios
2. Solve right triangles using the six trig. ratios
3. Know the ratios of the sides of the 30-60-90 special right triangle
4. Know the ratios of the sides of the 45-45-90 special right triangle
5. Apply the ratios of the special right triangles to real life application questions.
Next let’s put our angles inside triangles…specifically right triangles.
 Three Basic Trigonometric Ratios
sin 

side opposite 
side adjacent to 
side opposite 
, cos 
, tan 
hypotenuse
hypotenuse
side adjacent to 
o Remember the ratios are used on an acute angle.
o
We memorized
.
 The Reciprocal Ratios:
csc 




1
,
sec 
sin 
1
and
cot 
cos 
1
tan

Example 1: Using the triangle at the right, find all six
trigonometric functions of the angle .
29

21
Example 2: Given tan  =
5
12
, find the remaining trigonometric functions.
4a–1
Special Right Triangles:
A
#1: 45 – 45 – 90 Right Triangle (MEMORIZE THESE RATIOS)
a) If AC = 1, and m A = 45
, solve for the remaining parts of the triangle.
B
C
b) Find the sine, cosine and tangent values of 45.
A
#2: 30 -60 – 90 Right Triangle (MEMORIZE THESE RATIOS)
a) D ABC is equilateral, so each angle is
.
b) Draw the altitude of the triangle from A to BC .
Call the point of intersection D.
c) Therefore, m BAD =
.
d) Suppose AB = 2. Solve for the remaining parts of D BAD .
B
C
e) Find the sine, cosine and tangent values of 30and 60.
Example 4: A ladder is extended to reach the top floor of an 84 foot tall burning building. The fire fighters see
someone who needs rescuing in a window 8 feet below the roof. How far should the ladder be extended to
reach the roof if the ladder must be placed at the optimum operating angle of 60?
4a–2
6.4 Inverse Trigonometric Functions
Pre-Calculus
Learning Targets
1. Use the appropriate notation for inverse trigonometric functions.
2. Graph the inverse Sine, Cosine and Tangent functions.
3. List the correct Domain and Range of the inverse functions.
4. Find an exact solution to an expression involving an inverse sine, cosine or tangent.
5. Find the composition of trig functions and their inverses.
Inverse Sine
Inverse Cosine
Inverse Tangent
y sin1 x
y cos1 x
y tan1 x
y arcsin x
y arccos x
y arctan x
D: [–1, 1]
D: [–1, 1]
D: (–∞, ∞)
R: [0, 𝜋]
R: (− 2 , 2 )
𝜋 𝜋
R: [− 2 , 2 ]
𝜋 𝜋












Example 1: Find the exact value (in radians).
a) cos-1 0
b) sin-1 0
1




e) 𝑐𝑜𝑠 −1 (2)
f) 𝑡𝑎𝑛−1 − √3

c) arcsin
1
4b–3
d ) arctan 1
√2
g) 𝑎𝑟𝑐𝑠𝑖𝑛 (−

√3
)
2
h) 𝑎𝑟𝑐𝑐𝑜𝑠 −
√3
2
 Work from the inside out.
 Remember domain and range restrictions.
Example 2: Evaluate each expression.
a) sin⁡[arctan(−√3)]
b) 𝑐𝑜𝑠 −1 𝑐𝑜𝑠
3𝜋
2











Example 3: Find the algebraic expression equivalent to the given expression.
a) sin (cos1 x)
b) cot (sin1 2x)
Refer to pages 464 -465 Examples 4 and 5
4b–4
Solving Problems with Trigonometry
Pre-Calculus
Learning Targets
1. Set up and solve application problems involving right triangle trigonometry.
2. Use overlapping right triangles to solve word problems including the use of indirect measurement.
3. Solve problems involving simple harmonic motion.
Next we will apply what we know about the trigonometric functions, and their inverses, to solve real world application
problems. The most important part of these types of questions is an accurate, detailed picture.
Angle of Elevation: The acute angle measured from a horizontal
line UP to an object.
Angle of Depression: The acute angle measured from a horizontal
line DOWN to an object.
Angle of Depression
Angle of Elevation
Example 1: From a boat on the lake, the angle of elevation to the top of a cliff is 2042. If the base of the cliff is 1394
feet from the boat, how high is the cliff (to the nearest foot)?
4b–5