Do Now:Find all 6 Trig ratios:

A)

Solve for the missing sides:

θ32

16

7

30 3245

7

Inverse Trigonometric Functions

and Right Triangles

Inverse Trigonometric Functions and Right Triangles:

Using the calculator

Calculator must be in degree mode!

Finding Values:

sin(30°) select the SIN function on the calculator and enter 30° sin(30°) =

cos(40°) select the COS function on the calculator and enter 30° cos(40°) =

tan(15°) select the TAN function on the calculator and enter 30° tan(15°) =

csc(25°) 1

sin(25°)csc(25°) =

sec(32°) 1

cos(32°)sec(32°) =

cot(49°) 1

tan(49°)cot(49°) =

Using Trig to find missing side

Using Trig to find missing side

Using Trig to find missing side

Inverse

X2 = 4

X3 = 27

Using the right triangle, find the following:

(i) sin(θ)

(ii) cos(θ)

(iii) tan(θ)

1512

9

θ

What if we want to find the angle θ, given the sides of a triangle? To do this we use our functions.

When we calculate the sin(θ) it gives us a ratio of 𝑥 =𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

When we calculate the sin-1( ), it takes the ratio 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒, and returns the

angle θ.

In other words, if sin(θ) = x, then sin-1( ) =

cos(θ) = x, then cos-1( ) =

tan(θ) = x, then tan-1( ) =

• If sin(θ) maps θ to x. Then sin-1(x) maps x back to θ

• If cos(θ) maps θ to x. Then cos-1(x) maps x back to θ

• If tan(θ) maps θ to x. Then tan-1(x) maps x back to θ

Sine --> Inverse Sine Function:

(i) sin(θ) = 5

16

(ii) cos(θ) = 2

16

(iii) tan(θ) = 5

2

Using the Calculator to find Inverse Trig Values

sin-1(x) press 2nd then SIN and enter the x value

θ = sin-1( 2

3) θ = sin-1( 0.45) θ=sin-1(

3

7)

cos-1(x) press 2nd then COS and enter the x value

θ = cos-1(4

5) θ = cos-1(0.360) θ = cos-1(

4

7)

tan1(x) press 2nd, then TAN and enter the x value

θ = tan-1(9

15) θ = tan-1(

13

5) θ = tan-1(0.63)

To find Inverses of Sine, Cosine, and Tangent using our calculator:

sin-1(x) press 2nd then SIN and enter the x value

cos-1(x) press 2nd then COS and enter the x value

tan1(x) press 2nd, then TAN and enter the x value

Using the same right triangle, find the angle θ:

sin(θ) = 5

16 θ = sin-1(

5

16)= cos(θ) =

2

16 θ = cos-1(

2

16)

tan(θ) = 5

2 θ = tan-1(

5

2)=

θ

2

165

First find:

sin(θ ) = cos(θ) =

tan(θ) =

Then find the measure of angle θ. What do you notice?

θ

8

6

10

β

θ

θ

β

7

2524

6516

63

Practice:

1: Find each angle measure to the nearest degree.

a) sin(B) = 0.4848 b) cos(A) = 0.7431 c) tan(Q) = 0.5317

B = 29° A = 42° Q = 28°

e) cos(P) = 0.5878 f) cos(R) = 0.6157 g) tan(J) = 19.0811

P = 54.2° R = 52° J = 87°

h) cos(A) = 0.4226 j) sin(C) = 0.5150 k) tan(U) = 0.1410

A = 65° C = 31° U = 8.03°

Right Triangles:

a) 𝜃 = 𝑡𝑎𝑛−127

38= 35.4° b) 𝜃 = 𝑠𝑖𝑛−1

40

42= 72.2°

c) 𝜃 = 𝑠𝑖𝑛−111

27= 24° d) 𝜃 = 𝑠𝑖𝑛−1

12

24= 30°

e) 𝜃 = 𝑐𝑜𝑠−148

50= 16.3° f) 𝜃 = 𝑐𝑜𝑠−1

68

85= 36.9°

g) 𝜃 = 𝑠𝑖𝑛−16

8= 48.6°

h) 𝜃 = 𝑠𝑖𝑛−116

34= 28.1° ; 𝑐𝑜𝑠−1

30

34= 28.1° ; 𝑡𝑎𝑛−1

16

30= 28.1°

i) 𝜃 = 𝑠𝑖𝑛−142

70= 36.9° ; 𝑐𝑜𝑠−1

56

70= 36.9° ; 𝑡𝑎𝑛−1

42

56= 36.9°

j) 𝜃 = 𝑠𝑖𝑛−113

20= 40.5°

k) 𝜃 = 𝑠𝑖𝑛−124

25= 𝑐𝑜𝑠−1

7

25= 𝑡𝑎𝑛−1

24

7= 73.7°

l) 𝜃 = 𝑠𝑖𝑛−110

39= 14.9°

3: Find X

tan 𝑥 =300

400- 𝑥 = 𝑡𝑎𝑛−1

300

400= 36.9°

4: Draw and label all three sides of a right triangle that has a 40° angle and a hypotenuse of 10 cm.

5: Draw and label all three sides of a right triangle that has a 47° angle and a hypotenuse of 24 cm.

Closure:

1)When we calculate the sin(θ) it gives us a ratio of 𝑥 =𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒.

When we calculate the sin-1(x) it takes our ratio 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒, and gives us our angle

θ.

2) We can use our inverse trigonometric functions to calculate the measure of the angles of a right triangle